Question

Determine a soma dos termos da PG em que A5 = 12005 e Q = 7

Answer 1

Antes de podermos aplicar a equação da soma de termos da PG (abaixo), precisamos determinar o valor do 1° termo dessa PG.

$S_n~=~\dfrac{a_1\,.\,\left(q^n-1\right)}{q-1}$

Podemos determinar a₁ utilizando a equação do termo geral da PG:

$a_n~=~a_1~.~q^{n-1}\\\\\\a_5~=~a_1~.~q^{5-1}\\\\\\12005~=~a_1~.~7^4\\\\\\a_1~=~\frac{12005}{7^4}\\\\\\a_1~=~\frac{12005}{2401}\\\\\\\boxed{a_1~=~5}$

Agora sim podemos determinar a soma dos 5 termos dessa PG:

$S_5~=~\dfrac{5\,.\,\left(7^5-1\right)}{7-1}\\\\\\\\S_5~=~\dfrac{5\,.\,\left(16807-1\right)}{6}\\\\\\\\S_5~=~\dfrac{5\,.\,\left(16806\right)}{6}\\\\\\\\S_5~=~\dfrac{5~.~(2801)}{1}\\\\\\\\\boxed{S_5~=~14005}$