Question

Quantos termos existem na P.G. (5, 10, 20, ..., 10 240) ?

Answer 1

Vamos começar então determinando a razão (q) da PG.

$\boxed{\sf q~=~\dfrac{a_n}{a_{n-1}}}\\\\\\\sf Utilizando~os~dois~primeiros~termos~(a_1~e~a_2),~temos:\\\\\\q~=~\dfrac{a_2}{a_1}\\\\\\q~=~\dfrac{10}{5}\\\\\\\boxed{\sf q~=~2}$

Agora, utilizando a relação do termo geral da PG, podemos determinar o número "n" de termos

$\sf Termo~geral:~~\boxed{\sf a_n~=~a_m\cdot q^{n-m}}$

$\sf 10240~=~5\cdot 2^{n-1}\\\\\\\dfrac{10240}{5}~=~2^{n-1}\\\\\\2048~=~2^{n-1}\\\\\\2^{11}~=~2^{n-1}\\\\\\\backslash\!\!\! 2^{11}~=~\backslash\!\!\! 2^{n-1}\\\\\\11~=~n-1\\\\\\n~=~11+1\\\\\\\boxed{\sf n~=~12~termos}$

$\Huge{\begin{array}{c}\Delta \tt{\!\!\!\!\!\!\,\,o}\!\!\!\!\!\!\!\!\:\,\perp\end{array}}Qualquer~d\acute{u}vida,~deixe~ um~coment\acute{a}rio$