Question

A soma dos cinco primeiros termos da P.G. (2, -6, 18, ...) é :

Answer 1

$\textsf{Teorema para a soma dos n termos de uma P.G:}\\\\\\\fbox{ \mathsf{S_n=\dfrac{(a_n\cdot q)-a_1}{q-1}} }\\\\\\\textsf{F\'ormula de recorr\^encia de uma P.G: (irei usa-la para descobrir a raz\~ao)}\\\\\\\mathsf{a_1\cdot q=a_2}\\\\\mathsf{2\cdot q=-6}\\\\\mathsf{q= -\dfrac{6}{2}}$


$\mathsf{Teorema~para~o~termo~geral:~~\fbox{ \mathsf{a_n=a_1\cdot q^{n-1}} }}\\\\\\\textsf{Primeiro irei calcular o quinto termo (pelo teorema do termo geral) e}\\\textsf{depois irei descobrir a soma dos n termos.}\\\\\\\mathsf{a_5=2\cdot (-\dfrac{6}{2})^{5-1}~\Leftrightarrow~2\cdot(-\dfrac{6}{2})^4~\Leftrightarrow~2\cdot(-\dfrac{6^4}{2^4})~\Leftrightarrow~2\cdot(\dfrac{1296}{16})}\\\\\\\mathsf{a_5=2\cdot81}\\\\\\\mathsf{a_5=162}$


$\textsf{Substituindo os dados no primeiro teorema: (teorema da soma)}\\\\\\\mathsf{S_5=\dfrac{(162\cdot(-\dfrac{6}{2})-2}{-\dfrac{6}{2}-1}}\\\\\\\mathsf{S_5=\dfrac{-486-2}{-4}}\\\\\\\mathsf{S_5=\dfrac{-488}{-4}}\\\\\\\fbox{ \mathsf{S_5=122} }~~~~\checkmark$


$\textsf{Portanto a soma dos cinco primeiros termos dessa P.G \'e igual a 122}$

Answer 2

resolução!

q = a2 / a1

q = - 6/2

q = - 3

Sn = a1 ( q^n - 1 ) / q - 1

Sn = 2 ( - 3^5 - 1 ) / - 3 - 1

Sn = 2 ( - 243 - 1 ) / - 4

Sn = 2 * - 244 / - 4

Sn = - 488 / - 4

Sn = 122