Question

Em um PG a₆ = 64 e a₉ = 216. Qual o primeiro termo e a razão?

Answer 1

Vamos começar determinando a razão (q) utilizando a relação do termo geral para dois termos quaisquer.

$\boxed{a_n~=~a_m\cdot q^{n-m}}$

Substituindo os dados fornecidos, temos:

$a_9~=~a_6\cdot q^{9-6}\\\\\\216~=~64\cdot q^{3}\\\\\\q^3~=~\dfrac{216}{64}\\\\\\q^3~=~\dfrac{27}{8}\\\\\\q~=~\sqrt[3]{\dfrac{27}{8}}\\\\\\q~=~\dfrac{\sqrt[3]{27}}{\sqrt[3]{8}}\\\\\\\boxed{q~=~\frac{3}{2}~~ou~~ 1,5}$

Agora, utilizando novamente a relação do termo geral, podemos determinar o valor de a₁:

$a_6~=~a_1\cdot q^{6-1}\\\\\\64~=~a_1\cdot \left(\dfrac{3}{2}\right)^5\\\\\\64~=~a_1\cdot \dfrac{3^5}{2^5}\\\\\\64~=~a_1\cdot \dfrac{243}{32}\\\\\\a_1~=~64\cdot \dfrac{32}{343}\\\\\\\boxed{a_1~=~\dfrac{2048}{343}}\\\\\\\\\Huge{\begin{array}{c}\Delta \tt{\!\!\!\!\!\!\,\,o}\!\!\!\!\!\!\!\!\:\,\perp\end{array}}Qualquer~d\acute{u}vida,~deixe~ um~coment\acute{a}rio$