1MS.2 PROGRESSIONS ARITM I GEOM 1MS.2.10DT Progressió Aritmètica

Progressió aritmètica

En una progressió aritmètica, cada terme es dedueix de l'anterior sumant una quantitat fixa anomenada diferència:

a_n = a_n-1 + d

El terme general és a_n = a₁ + (n-1)d
I la suma dels n primers termes es calcula amb:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»S«/mi»«mi mathvariant=¨bold¨»n«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mfrac mathcolor=¨#003300¨»«mrow»«mi mathvariant=¨bold¨»n«/mi»«mo mathvariant=¨bold¨»(«/mo»«msub»«mi mathvariant=¨bold¨»a«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»+«/mo»«msub»«mi mathvariant=¨bold¨»a«/mi»«mi mathvariant=¨bold¨»n«/mi»«/msub»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«/mrow»«/mstyle»«/math»

]]> 0.0000000 0.0000000 0 1MS.2.11Q PA a1 i d En la progressió aritmètica a_n,«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo mathcolor=¨#003300¨»§#160;«/mo»«msub mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»i_«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»t_«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»i«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«msub mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»i_«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»t_«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«/mstyle»«/math».

Calcula el primer terme: {#1}

i la diferència: {#2}

]]> 2.0000000 0.5000000 0 <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>i_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>15</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>i_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>15</mn><mo>)</mo></mtd></mtr></mtable><mrow><mi>i_2</mi><mo>></mo><mi>i_1</mi><mo>+</mo><mn>3</mn></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>10</mn><mo>,</mo><mn>10</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>7</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>a_1</mi><mo>+</mo><mo>(</mo><mi>x</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>*</mo><mi>d</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t_1</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>i_1</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t_2</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>i_2</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>=</mo><mi>i_2</mi><mo>-</mo><mi>i_1</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mi>t_2</mi><mo>-</mo><mi>t_1</mi></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>28</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>52</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer></correctAnswer></correctAnswers><options><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="casSession"></data></localData></question> Entre els dos termes, hi ha #i_2 - #i_1 = #i diferències que valen #D. Es pot doncs trobar quant val una diferència, i després el primer terme.

]]> 1MS.2.12Q PA a1 i ai sabent 2 termes En la progressió aritmètica a_n, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msub mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»i_«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»t_«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»i«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«msub mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»i_«/mi»«/mrow»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»t_«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«/mstyle»«/math».

Calcula a₁: {#1}

Calcula a_{#i_3}: {#2}

]]> 2.0000000 0.3333333 0 <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>i_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>15</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>i_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>15</mn><mo>)</mo></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><mi>i_2</mi><mo>></mo><mi>i_1</mi><mo>+</mo><mn>3</mn></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>5</mn><mo>,</mo><mn>11</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i_3</mi><mo>=</mo><mi>i_2</mi><mo>+</mo><mi>m</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>10</mn><mo>,</mo><mn>10</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>7</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>a_1</mi><mo>+</mo><mo>(</mo><mi>x</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>*</mo><mi>d</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t_1</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>i_1</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t_2</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>i_2</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>=</mo><mi>i_2</mi><mo>-</mo><mi>i_1</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mi>t_2</mi><mo>-</mo><mi>t_1</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_3</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>i_3</mi><mo>)</mo></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>28</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>52</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer></correctAnswer></correctAnswers><options><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="casSession"></data></localData></question> Entre els dos termes, hi ha #i_2 - #i_1 = #i diferències que valen #D. Es pot doncs trobar quant val una diferència, i després el primer terme.

a_{#i_1} = #t_1 està a #k diferències de a₁; per a trobar a₁ cal fer: #t_1 - #k · d.

Després es calcula a_{#i_3} amb la fórmula: a_n = a₁ + (n-1)d, amb a1 = #a_1, n = #i_3 i d = #d

]]> 1MS.2.15Q PA a1 + d → 2 termes En una progressió aritmètica, el primer terme és «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«/mrow»«/mstyle»«/math» i la diferència «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»d«/mi»«/mrow»«/mstyle»«/math».

Calcula

a) «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«msub mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«/mrow»«/mstyle»«/math»

b) «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«msub mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/msub»«/mstyle»«/math»

]]> 1.0000000 0.5000000 0 0 a) =#sol1b) =#sol2]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>12</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>d</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>12</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>n1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>6</mn><mo>,</mo><mn>12</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>n2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>12</mn><mo>,</mo><mn>19</mn><mo>)</mo></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><mi>a1</mi><mo>≠</mo><mi>d</mi></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><mi>a1</mi><mo>+</mo><mfenced><mrow><mi>n1</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>*</mo><mi>d</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mi>a1</mi><mo>+</mo><mfenced><mrow><mi>n2</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>*</mo><mi>d</mi></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>7</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n1</mi><mo>,</mo><mi>n2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo>,</mo><mn>18</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>65</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>137</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>b</mi><mo>)</mo><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputCompound">true</data><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> N'hi ha prou amb substituir n per l'índex en:

a) «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«msub mathcolor=¨#00007F¨»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»a«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»§#160;«/mo»«/mrow»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»a«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#00007F¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»+«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mfenced mathcolor=¨#00007F¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»-«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#183;«/mo»«mfenced mathcolor=¨#00007F¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«/mrow»«/mfenced»«mspace linebreak=¨newline¨/»«/mstyle»«/math»

]]> N'hi ha prou amb substituir n per l'índex en:

b) «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«msub mathcolor=¨#00007F¨»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»a«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#160;«/mo»«/mrow»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»a«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#00007F¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»+«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mfenced mathcolor=¨#00007F¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»-«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#183;«/mo»«mfenced mathcolor=¨#00007F¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«/mrow»«/mfenced»«mspace linebreak=¨newline¨/»«/mstyle»«/math»

]]> 1MS.2.16Q PA 2Termes → d En una progressió aritmètica,

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«msub mathcolor=¨#003300¨»«mi mathvariant=¨bold-italic¨ mathcolor=¨#003300¨»a«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#003300¨»s«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#003300¨»o«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#003300¨»l«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#003300¨»i«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«msub mathcolor=¨#003300¨»«mi mathvariant=¨bold-italic¨ mathcolor=¨#003300¨»a«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#003300¨»s«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#003300¨»o«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#003300¨»l«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«/mrow»«/mstyle»«/math»

Quina és la diferència?

]]> 1.0000000 0.5000000 0 0 #d]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>12</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>d</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>12</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>n1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>6</mn><mo>,</mo><mn>12</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>n2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>15</mn><mo>,</mo><mn>19</mn><mo>)</mo></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><mi>a1</mi><mo>≠</mo><mi>d</mi></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><mi>a1</mi><mo>+</mo><mfenced><mrow><mi>n1</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>*</mo><mi>d</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mi>a1</mi><mo>+</mo><mfenced><mrow><mi>n2</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>*</mo><mi>d</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m3</mi><mo>=</mo><mi>sol2</mi><mo>-</mo><mi>sol1</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n3</mi><mo>=</mo><mi>n2</mi><mo>-</mo><mi>n1</mi></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>4</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n1</mi><mo>,</mo><mi>n2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>,</mo><mn>13</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>26</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>50</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>24</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n3</mi></math></input></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>#</mo><mi mathvariant="normal">d</mi><mspace linebreak="newline"/></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> La diferència entre els dos termes és «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»m«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#00007F¨»3«/mn»«/mrow»«/mstyle»«/math» i correspon a «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»n«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#00007F¨»3«/mn»«/mrow»«/mstyle»«/math» diferències

]]>

]]> 1MS.2.17Q PA 2Termes → ai En una progressió aritmètica,

Calcula «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«msub mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»s«/mi»«/mrow»«/msub»«/mstyle»«/math»

]]> 1.0000000 0.5000000 0 0 #sol]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>12</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>d</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>12</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>n1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>6</mn><mo>,</mo><mn>12</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>n2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>15</mn><mo>,</mo><mn>19</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>s</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>5</mn><mo>,</mo><mn>29</mn><mo>)</mo></mtd></mtr></mtable><mrow><mfenced><mrow><mi>a1</mi><mo>≠</mo><mi>d</mi></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>s</mi><mo>≠</mo><mi>n1</mi></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>s</mi><mo>≠</mo><mi>n2</mi></mrow></mfenced></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><mi>a1</mi><mo>+</mo><mfenced><mrow><mi>n1</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>*</mo><mi>d</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mi>a1</mi><mo>+</mo><mfenced><mrow><mi>n2</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>*</mo><mi>d</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m3</mi><mo>=</mo><mi>sol2</mi><mo>-</mo><mi>sol1</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n3</mi><mo>=</mo><mi>n2</mi><mo>-</mo><mi>n1</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi><mo>=</mo><mi>a1</mi><mo>+</mo><mfenced><mrow><mi>s</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>*</mo><mi>d</mi></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>4</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n1</mi><mo>,</mo><mi>n2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>,</mo><mn>13</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>26</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>50</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>24</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n3</mi></math></input></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mspace linebreak="newline"/></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> La diferència entre els dos termes és «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»m«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#00007F¨»3«/mn»«/mrow»«/mstyle»«/math» i correspon a «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»n«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#00007F¨»3«/mn»«/mrow»«/mstyle»«/math» diferències.

Trobada la diferència #d només cal sumar o restar diferències a «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«msub mathcolor=¨#00007F¨»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»a«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/msub»«/mstyle»«/math»

]]>

]]> 1MS.2.18Q PA 2Termes → d i a1 En un progressió aritmètica, el #p è terme és #a_p i el #q è és #a_q. A partir d'aquestes dades, dedueix el valor:

a) de la diferència

b) del primer terme a₁

]]> 1.0000000 0.5000000 0 0 a) =#sol1b) =#sol2]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>5</mn><mo>,</mo><mn>8</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>12</mn><mo>,</mo><mn>19</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>12</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>a_1</mi><mo>+</mo><mo>(</mo><mi>n</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>*</mo><mi>d</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_p</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>p</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_q</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>q</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mi>a_q</mi><mo>-</mo><mi>a_p</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><mi>q</mi><mo>-</mo><mi>p</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><mi>d</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mi>a_1</mi></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_1</mi><mo>,</mo><mi>d</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>,</mo><mn>11</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>11</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>b</mi><mo>)</mo><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="inputCompound">true</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> N'hi ha prou amb pensar que entre #a_p i #a_q = #u s'han afegit #q - #p = #v diferències.

]]> 1MS.2.31Q PA an i Sn En un progressió aritmètica, el primer terme és #a_1 i la diferència #d. A partir d'aquestes dades, deduïu el valor:

del terme a_#n = {#1}

de la suma dels #n primers termes: Sn = {#2}

]]> 2.0000000 0.5000000 0 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>12</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>11</mn><mo>,</mo><mn>29</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>a_1</mi><mo>+</mo><mo>(</mo><mi>n</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>*</mo><mi>d</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>=</mo><mfrac><mrow><mo>(</mo><mi>a_1</mi><mo>+</mo><mi>a</mi><mo>)</mo><mo>*</mo><mi>n</mi></mrow><mn>2</mn></mfrac></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>22</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>150</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1683</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession></question> Només cal aplicar les expressions teòriques de a_n i S_n

]]> 1MS.2.32Q PA a1 i Sn En la progressió aritmètica a_n, a_{#i_1} = #t_1 i a_{#i_2} = #t_2.

Calcula a₁: {#1}

Calcula la suma dels #i_3 primer termes: S_{#i_3}: {#2}

]]>

]]> 2.0000000 0.3333333 0 <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>i_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>15</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>i_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>15</mn><mo>)</mo></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><mi>i_2</mi><mo>></mo><mi>i_1</mi><mo>+</mo><mn>3</mn></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>5</mn><mo>,</mo><mn>11</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i_3</mi><mo>=</mo><mi>i_2</mi><mo>+</mo><mi>m</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>10</mn><mo>,</mo><mn>10</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>7</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>a_1</mi><mo>+</mo><mo>(</mo><mi>x</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>*</mo><mi>d</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t_1</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>i_1</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t_2</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>i_2</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>=</mo><mi>i_2</mi><mo>-</mo><mi>i_1</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mi>t_2</mi><mo>-</mo><mi>t_1</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_3</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>i_3</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>=</mo><mfrac><mrow><mi>i_3</mi><mo>*</mo><mfenced><mrow><mi>a_1</mi><mo>+</mo><mi>a_3</mi></mrow></mfenced></mrow><mn>2</mn></mfrac></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>28</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>52</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer></correctAnswer></correctAnswers><options><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="casSession"></data></localData></question> Entre els dos termes, hi ha #i_2 - #i_1 = #i diferències que valen #D. Es pot doncs trobar quant val una diferència, i després el primer terme.

a_{#i_1} = #t_1 està a #k diferències de a₁; per a trobar a₁ cal fer: #t_1 - #k · d.

Després es calcula la suma amb la fórmula: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»S«/mi»«mi mathvariant=¨bold¨»n«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mi mathvariant=¨bold¨»n«/mi»«mo mathvariant=¨bold¨»(«/mo»«msub»«mi mathvariant=¨bold¨»a«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»+«/mo»«msub»«mi mathvariant=¨bold¨»a«/mi»«mi mathvariant=¨bold¨»n«/mi»«/msub»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«/mstyle»«/math»

amb a₁ = #a_1, n = #i_3 i d = #d

]]> 1MS.2.35Q PA a1+d → an i Sn En un progressió aritmètica, el primer terme és «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a_«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«/mrow»«/mstyle»«/math» i la diferència «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»d«/mi»«/mrow»«/mstyle»«/math». A partir d'aquestes dades, deduïu el valor:

a) del terme «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«msub mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«/mrow»«/msub»«/mstyle»«/math»

]]> 1.0000000 0.5000000 0 0 a) =#solb) =#sol2]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>10</mn><mo>,</mo><mn>20</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>a_1</mi><mo>+</mo><mo>(</mo><mi>x</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>*</mo><mi>d</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mfrac><mrow><mi>n</mi><mo>*</mo><mfenced><mrow><mi>a_1</mi><mo>+</mo><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mfenced></mrow><mn>2</mn></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_1</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>n</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>,</mo><mn>7</mn><mo>,</mo><mn>10</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>68</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>365</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mspace linebreak="newline"/><mi>b</mi><mo>)</mo><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputCompound">true</data><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> Només cal aplicar les fórmules

]]> 1MS.2.41 PA: falcons

En un escala dels falcons, en cada pis hi ha una persona menys que en el pis de sota. A sobre la última fila on hi ha una persona, és on s'enfila l'enxaneta.
Quantes persones hi ha en una escala de #a (pisos)?

Resposta = {#1}

]]> 1.0000000 0.5000000 0 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>7</mn><mo>,</mo><mn>12</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mfrac><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>a</mi><mo>)</mo><mo>*</mo><mi>a</mi></mrow><mn>2</mn></mfrac><mo>+</mo><mn>1</mn></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>37</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession></question> és una progressió aritmètica de diferència (-1) i de primer terme #a. Cal sumar l'enxaneta!

]]> 1MS.2.50DT Progressió Geomètrica

Progressió geomètrica

En una progressió geomètrica, cada terme es dedueix de l'anterior multiplicant per una quantitat fixa anomenada raó:

a_n = a_n-1 + d

El terme general és a_n = a₁ · r^n-1
El producte dels n primers termes és: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»P«/mi»«mi mathvariant=¨bold¨»n«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«msqrt mathcolor=¨#003300¨»«msup»«mfenced»«mrow»«msub»«mi mathvariant=¨bold¨»a«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»a«/mi»«mi mathvariant=¨bold¨»n«/mi»«/msub»«/mrow»«/mfenced»«mi mathvariant=¨bold¨»n«/mi»«/msup»«/msqrt»«/mrow»«/mstyle»«/math»
La suma dels n primers termes és: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»S«/mi»«mi mathvariant=¨bold¨»n«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mfrac mathcolor=¨#003300¨»«mrow»«msub»«mi mathvariant=¨bold¨»a«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»(«/mo»«msup»«mi mathvariant=¨bold¨»r«/mi»«mi mathvariant=¨bold¨»n«/mi»«/msup»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«mrow»«mi mathvariant=¨bold¨»r«/mi»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math»

]]> 0.0000000 0.0000000 0 1MS.2.51Q PG 2Termes → r En una progressió geomètrica, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«msub mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»§#160;«/mo»«/mrow»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»sol«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»i«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«msub mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#160;«/mo»«/mrow»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»sol«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«/mstyle»«/math»

Calcula la raó.

]]> 1.0000000 0.5000000 0 0 #r <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>r</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>n1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>n2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>7</mn><mo>,</mo><mn>12</mn><mo>)</mo></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><mi>a1</mi><mo>≠</mo><mi>d</mi></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><mi>a1</mi><mo>*</mo><msup><mi>r</mi><mrow><mi>n1</mi><mo>-</mo><mn>1</mn></mrow></msup></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mi>a1</mi><mo>*</mo><msup><mi>r</mi><mrow><mi>n2</mi><mo>-</mo><mn>1</mn></mrow></msup></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n3</mi><mo>=</mo><mi>n2</mi><mo>-</mo><mi>n1</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m3</mi><mo>=</mo><mfrac><mi>sol2</mi><mi>sol1</mi></mfrac></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n1</mi><mo>,</mo><mi>n2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>,</mo><mn>11</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>36</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>236196</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer>#r</correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> Per passar de «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«msub mathcolor=¨#00007F¨»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»a«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«/mrow»«/msub»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«msub mathcolor=¨#00007F¨»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»a«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/msub»«/mrow»«/mstyle»«/math» s'ha multiplicat:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«msub mathcolor=¨#00007F¨»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»a«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»per«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»l«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»r«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»a«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»§#243;«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»e«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»l«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»e«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»v«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»a«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»d«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»a«/mi»«mfenced mathcolor=¨#00007F¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»-«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«/mstyle»«/math»
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«msub mathcolor=¨#00007F¨»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»a«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»§#160;«/mo»«/mrow»«/msub»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»p«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»e«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»r«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mfrac mathcolor=¨#00007F¨»«msub»«mi mathvariant=¨bold¨»a«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/msub»«msub»«mi mathvariant=¨bold¨»a«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/msub»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»=«/mo»«mfrac mathcolor=¨#00007F¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»s«/mi»«mi mathvariant=¨bold¨»o«/mi»«mi mathvariant=¨bold¨»l«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»s«/mi»«mi mathvariant=¨bold¨»o«/mi»«mi mathvariant=¨bold¨»l«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfrac»«/mstyle»«/math»

i es pot deduir la raó, ja que «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«msup mathcolor=¨#00007F¨»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»r«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»3«/mn»«/mrow»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»m«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#00007F¨»3«/mn»«/mstyle»«/math»

]]>

]]> 1MS.2.52Q PG 2Termes → a1 i r En una progressió geomètrica, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«msub mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»§#160;«/mo»«/mrow»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»sol«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»i«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«msub mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#160;«/mo»«/mrow»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»sol«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«/mstyle»«/math»

a) Calcula la raó.

b) Calcula el primer terme

]]> 1.0000000 0.3333333 0 0 a) =#rb) =#a1]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>r</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>n1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>n2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>7</mn><mo>,</mo><mn>12</mn><mo>)</mo></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><mi>a1</mi><mo>≠</mo><mi>d</mi></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><mi>a1</mi><mo>*</mo><msup><mi>r</mi><mrow><mi>n1</mi><mo>-</mo><mn>1</mn></mrow></msup></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mi>a1</mi><mo>*</mo><msup><mi>r</mi><mrow><mi>n2</mi><mo>-</mo><mn>1</mn></mrow></msup></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n3</mi><mo>=</mo><mi>n2</mi><mo>-</mo><mi>n1</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m3</mi><mo>=</mo><mfrac><mi>sol2</mi><mi>sol1</mi></mfrac></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n1</mi><mo>,</mo><mi>n2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>,</mo><mn>11</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>36</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>236196</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo> </mo><mo>=</mo><mo>#</mo><mi>r</mi><mspace linebreak="newline"/><mi>b</mi><mo>)</mo><mo> </mo><mo>=</mo><mo>#</mo><mi>a</mi><mn>1</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/><data name="inputCompound">true</data></localData></question> Per passar de «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«msub mathcolor=¨#00007F¨»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»a«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«/mrow»«/msub»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«msub mathcolor=¨#00007F¨»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»a«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/msub»«/mrow»«/mstyle»«/math» s'ha multiplicat:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«msub mathcolor=¨#00007F¨»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»a«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»per«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»l«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»r«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»a«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»§#243;«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»e«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»l«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»e«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»v«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»a«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»d«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»a«/mi»«mfenced mathcolor=¨#00007F¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»-«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«/mstyle»«/math»
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«msub mathcolor=¨#00007F¨»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»a«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»§#160;«/mo»«/mrow»«/msub»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»p«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»e«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»r«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mfrac mathcolor=¨#00007F¨»«msub»«mi mathvariant=¨bold¨»a«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/msub»«msub»«mi mathvariant=¨bold¨»a«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/msub»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»=«/mo»«mfrac mathcolor=¨#00007F¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»s«/mi»«mi mathvariant=¨bold¨»o«/mi»«mi mathvariant=¨bold¨»l«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»s«/mi»«mi mathvariant=¨bold¨»o«/mi»«mi mathvariant=¨bold¨»l«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfrac»«/mstyle»«/math»

]]> Coneguda la raó, es pot calcular a₁dividint «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«msub mathcolor=¨#00007F¨»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»a«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»§#160;«/mo»«/mrow»«/msub»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»per«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»la«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»ra§#243;«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»elevada«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mfenced mathcolor=¨#00007F¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»-«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«/mrow»«/mstyle»«/math»

]]> 1MS.2.55Q PG 2Termes → a1 i r En la progressió aritmètica a_n, a_{#i_1} = #t_1 i a_{#i_2} = #t_2.

Calcula el primer terme: {#1}

i la raó: {#2}

]]> 2.0000000 0.5000000 0 <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>i_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>9</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>i_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>9</mn><mo>)</mo></mtd></mtr></mtable><mrow><mi>i_2</mi><mo>></mo><mi>i_1</mi><mo>+</mo><mn>3</mn></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>6</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>7</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>a_1</mi><mo>*</mo><msup><mi>r</mi><mi>x</mi></msup></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t_1</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>i_1</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t_2</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>i_2</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>=</mo><mi>i_2</mi><mo>-</mo><mi>i_1</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mfrac><mi>t_2</mi><mi>t_1</mi></mfrac></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>28</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>52</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer></correctAnswer></correctAnswers><options><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="casSession"/></localData></question> Entre els dos termes, hi ha #i_2 - #i_1 = #i diferències que valen #D. Es pot doncs trobar quant val una diferència, i després el primer terme.

]]> 1MS.2.61Q PG a1+r → 2Termes En una progressió geomètrica, el primer terme és «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«/mrow»«/mstyle»«/math» i la raó «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#003300¨»r«/mi»«/mstyle»«/math».

Calcula

]]> 1.0000000 0.5000000 0 0 a) =#sol1b) =#sol2]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>r</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>n1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>n2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>7</mn><mo>,</mo><mn>12</mn><mo>)</mo></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><mi>a1</mi><mo>≠</mo><mi>d</mi></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><mi>a1</mi><mo>*</mo><msup><mi>r</mi><mrow><mi>n1</mi><mo>-</mo><mn>1</mn></mrow></msup></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mi>a1</mi><mo>*</mo><msup><mi>r</mi><mrow><mi>n2</mi><mo>-</mo><mn>1</mn></mrow></msup></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n1</mi><mo>,</mo><mi>n2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>,</mo><mn>11</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>18</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>118098</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>b</mi><mo>)</mo><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputCompound">true</data><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> N'hi ha prou amb substituir n per l'índex en:

a) «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«msub mathcolor=¨#00007F¨»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»a«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»§#160;«/mo»«/mrow»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»a«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#00007F¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#183;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»r«/mi»«msup mathcolor=¨#00007F¨»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»-«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«/msup»«mspace linebreak=¨newline¨/»«/mstyle»«/math»

]]> N'hi ha prou amb substituir n per l'índex en:

b) «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«msub mathcolor=¨#00007F¨»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»a«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#160;«/mo»«/mrow»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»a«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#00007F¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#183;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»r«/mi»«msup mathcolor=¨#00007F¨»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»-«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«/msup»«mspace linebreak=¨newline¨/»«/mstyle»«/math»

]]> 1MS.2.71Q PG a1+r → an i Sn En un progressió geomètrica, el primer terme és «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a_«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«/mrow»«/mstyle»«/math» i la raó r = «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»d«/mi»«/mrow»«/mstyle»«/math». A partir d'aquestes dades, dedueix el valor:

b) de la suma dels «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»n«/mi»«/mrow»«/mstyle»«/math» primers termes: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«msub mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»S«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«/mrow»«/msub»«/mstyle»«/math»

]]> 1.0000000 0.5000000 0 0 a) =#ab) =#S]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>12</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>a_1</mi><mo>*</mo><msup><mi>d</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>=</mo><mfrac><mrow><mi>a_1</mi><mo>*</mo><mo>(</mo><msup><mi>d</mi><mi>n</mi></msup><mo>-</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>d</mi><mo>-</mo><mn>1</mn></mrow></mfrac></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_1</mi><mo>,</mo><mi>d</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>,</mo><mn>9</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>,</mo><mi>a</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>,</mo><mn>243</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>273</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo> </mo><mo>=</mo><mo>#</mo><mi>a</mi><mspace linebreak="newline"/><mi>b</mi><mo>)</mo><mo> </mo><mo>=</mo><mo>#</mo><mi>S</mi></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputCompound">true</data><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> Només cal aplicar les expressions teòriques de a_n i S_n

]]> 1MS.2.75Q PG: a1+r → an i Sn En un progressió geomètrica, el primer terme és #a_1 i la raó #d. A partir d'aquestes dades, deduïu el valor:

del terme a_#n = {#1}

de la suma dels #n primers termes: Sn = {#2}

]]> 2.0000000 0.5000000 0 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>12</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>a_1</mi><mo>*</mo><msup><mi>d</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>=</mo><mfrac><mrow><mi>a_1</mi><mo>*</mo><mo>(</mo><msup><mi>d</mi><mi>n</mi></msup><mo>-</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>d</mi><mo>-</mo><mn>1</mn></mrow></mfrac></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>472392</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>531440</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession></question> Aplica les expressions teòriques de a_n i S_n

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