Question

Determine o décimo terceiro termo da PG (2.187, 729, ...)

Answer 1

Vamos começar determinando a razão (q) da PG:

$razao~=~\frac{a_2}{a_1}\\\\\\razao~=~\frac{729}{2187}\\\\\\\boxed{razao~=~\frac{1}{3}}$

Agora, utilizando a equação do termo geral da PG, podemos determinar o valor do 13° termo:

$\boxed{a_n~=~a_1~.~q^{n-1}}\\\\\\a_{13}~=~2187~.~\left(\frac{1}{3}\right)^{13-1}\\\\\\a_{13}~=~2187~.~\left(\frac{1}{3}\right)^{12}\\\\\\a_{13}~=~3^{7}~.~\frac{1^{12}}{3^{12}}\\\\\\a_{13}~=~\frac{3^7~.~1}{3^{12}}\\\\\\a_{13}~=~\frac{1~.~1}{3^{12-7}}\\\\\\a_{13}~=~\frac{1}{3^5}\\\\\\\boxed{a_{a3}~=~\frac{1}{243}}$

Answer 2

resolução!

q = a2 / a1

q = 729 / 2187

q = 1 / 3

a13 = a1 * q^12

a13 = 2187 * (1/3)^12

a13 = 2187 * 1/531441

a13 = 2187/531441

a13 = 1/243