1rBTX 05. SUCCESSIONS/1.5.1 Definició i monotonia 1.5.1.34Q MonotoniaRacionalG2G1 Considera la successió a_n = «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_n«/mi»«/mrow»«/mstyle»«/math»

a) Calcula a_n+1
Format de la resposta «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mfrac»«mrow»«mo»(«/mo»«mi»n«/mi»«mo»+«/mo»«mn»1«/mn»«msup»«mo»)«/mo»«mn»2«/mn»«/msup»«/mrow»«mrow»«mn»2«/mn»«mi»n«/mi»«mo»+«/mo»«mn»3«/mn»«/mrow»«/mfrac»«/mstyle»«/math»

b) Calcula a_n+1 - a_n

c) És creixent la successió (S/N) ?

]]> 1.0000000 0.5000000 0 0 a) =#sol1b) =#sol2c) =#sol3]]> <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>b1</mi><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>b2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b3</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><naturalnumbers/></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><mfrac><mrow><mi>b1</mi><mo>*</mo><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mi>b3</mi></mrow><mrow><mn>2</mn><mo>*</mo><mi>n</mi><mo>+</mo><mi>b2</mi></mrow></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_n</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>sol1</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mi>sol1</mi><mo>-</mo><mi>a_n</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>b1</mi><mo>></mo><mn>0</mn></mrow><mtable><mtr><mtd><mi>sol3</mi><mo>=</mo><mi>S</mi></mtd></mtr><mtr><mtd><mi>t1</mi><mo>=</mo><mo>"</mo><mi>és</mi><mo>"</mo></mtd></mtr><mtr><mtd><mi>u1</mi><mo>=</mo><mo>"</mo><mi>és</mi><mo>"</mo></mtd></mtr></mtable><mtable><mtr><mtd><mi>sol3</mi><mo>=</mo><mi>N</mi></mtd></mtr><mtr><mtd><mi>t1</mi><mo>=</mo><mo>"</mo><mi>no</mi><mo> </mo><mi>és</mi><mo>"</mo></mtd></mtr><mtr><mtd><mi>u1</mi><mo>=</mo><mo>"</mo><mi>no</mi><mo> </mo><mi>és</mi><mo>"</mo></mtd></mtr></mtable></apply></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"/></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_n</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn></mrow><mrow><mn>2</mn><mo>*</mo><mi>n</mi><mo>+</mo><mn>5</mn></mrow></mfrac></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"><mfrac><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mo>*</mo><mi>n</mi><mo>-</mo><mn>3</mn></mrow><mrow><mn>2</mn><mo>*</mo><mi>n</mi><mo>+</mo><mn>7</mn></mrow></mfrac></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"><mfrac><mrow><mn>2</mn><mo>*</mo><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><mo>*</mo><mi>n</mi><mo>+</mo><mn>13</mn></mrow><mrow><mn>4</mn><mo>*</mo><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>24</mn><mo>*</mo><mi>n</mi><mo>+</mo><mn>35</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t1</mi><mo>,</mo><mi>u1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><ms>és</ms><mo>,</mo><ms>és</ms></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><mspace linebreak="newline"/><mi>c</mi><mo>)</mo><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>3</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> Per calcular a_n+1,has de substituir n per n+1:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»a«/mi»«mrow»«mi mathvariant=¨bold¨»n«/mi»«mo mathvariant=¨bold¨»+«/mo»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»b«/mi»«mn mathvariant=¨bold¨»1«/mn»«msup»«mfenced»«mrow»«mi mathvariant=¨bold¨»n«/mi»«mo mathvariant=¨bold¨»+«/mo»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»-«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»b«/mi»«mn mathvariant=¨bold¨»3«/mn»«/mrow»«mrow»«mn mathvariant=¨bold¨»2«/mn»«mfenced»«mrow»«mi mathvariant=¨bold¨»n«/mi»«mo mathvariant=¨bold¨»+«/mo»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«mo mathvariant=¨bold¨»+«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»b«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math»

Atenció: (n+1)² = n² + 2n + 1

Com que la diferència a_n+1 - a_n #t1 positiva, la successió #u1 creixent

]]>