Question

Seja a soma dos termos 2+5+8.......+X = 301. Calcule o valor de X
*

44
47
41
50

09 - Se em uma PA a3+a6 = 34 e a4 +a7 = 42. Calcule a1 e r respectivamente

1;2
2;3
3;4
4;5

10 - Qual a soma dos 14 primeiros termos da PA da questão anterior?


506
460
420
406

Answer 1

$\Large\boxed{\begin{array}{l}\sf 2+5+8+\dotsc+x=301\\\sf a_1=2~~a_n=x~S_n=301\\\sf S_n=\dfrac{n\cdot (a_1+a_n)}{2}\\\\\sf 301=\dfrac{n\cdot(2+x)}{2}\\\\\sf 2n+nx=602\\\sf a_n=a_1+(n-1)\cdot r\\\sf x=2+(n-1)\cdot3\\\sf x=2+3n-3\\\sf x=3n-1\\\sf 2n+n\cdot(3n-1)=602\\\sf 3n^2-n+2n-602=0\\\sf 3n^2+n-602=0\end{array}}$

$\Large\boxed{\begin{array}{l}\sf\Delta=b^2-4ac\\\sf\Delta=1^2-4\cdot3\cdot(-602)\\\sf\Delta=1+7224\\\sf\Delta=7225\\\sf n=\dfrac{-b\pm\sqrt{\Delta}}{2a}\\\\\sf n=\dfrac{-1\pm\sqrt{7225}}{2\cdot3}\\\\\sf n=\dfrac{-1\pm85}{6}\begin{cases}\sf n_1=\dfrac{-1+85}{6}=\dfrac{84}{6}=14\\\\\sf n_2=\dfrac{-1-85}{6}=\dfrac{-86\div2}{6\div2}=-\dfrac{43}{3}\end{cases}\\\\\sf como~n > 0\implies n=14\end{array}}$

$\Large\boxed{\begin{array}{l}\sf x=3n-1\\\sf x=3\cdot14-1\\\sf x=42-1\\\sf x=41\checkmark\\\huge\boxed{\boxed{\boxed{\boxed{\sf\maltese~alternativa~c}}}}\end{array}}$

$\Large\boxed{\begin{array}{l}\rm 09)\\\sf a_3+a_6=34\\\sf a_1+2r+a_1+5r=34\\\sf 2a_1+7r=34\\\sf a_4+a_7=42\\\sf a_1+3r+a_1+6r=42\\\sf 2a_1+9r=42\\\begin{cases}\sf 2a_1+7r=34\cdot(-1)\\\sf 2a_1+9r=42\end{cases}\\+\underline{\begin{cases}\sf -\diagdown\!\!\!\!\!\!\!2a_1-7r=-34\\\sf \diagdown\!\!\!\!\!\!\!2a_1+9r=42\end{cases}}\\\sf 2r=8\\\sf r=\dfrac{8}{2}\\\\\sf r=4\\\sf 2a_1+7r=34\\\sf 2a_1+7\cdot4=34\\\sf 2a_1+28=34\\\sf 2a_1=34-28\\\sf 2a_1=6\\\sf a_1=\dfrac{6}{2}\\\\\sf a_1=3\\\huge\boxed{\boxed{\boxed{\boxed{\sf\maltese~alternativa~c}}}}\end{array}}$

$\Large\boxed{\begin{array}{l}\rm 10)~~\sf a_1=3~r=4\\\sf a_{14}=a_1+13r\\\sf a_{14}=3+13\cdot 4\\\sf a_{14}=3+52\\\sf a_{14}=55\\\sf S_{14}=\dfrac{\diagdown\!\!\!\!\!\!\!14\cdot(3+55)}{\diagdown\!\!\!\!2}\\\\\sf S_{14}=7\cdot58\\\sf S_{14}=406\\\huge\boxed{\boxed{\boxed{\boxed{\sf\maltese~alternativa~d}}}}\end{array}}$