Question

quantos termos tem a P.G (2187,729,.....,1 )




Determine a9 na P.G em que a1=64 e q=1/2

Answer 1

Olá,

dos dados..

$\begin{cases}a_1=2.187\\ q= \dfrac{a_2}{a_1}= \dfrac{729}{2.187}= \dfrac{1}{3}\\ a_n=1\\ n=? \end{cases}$

Podemos usar a fórmula do termo geral da P.G.:

$a_n=a_1\cdot q^{n-1}\\\\ 1=2.187\cdot \dfrac{1}{3}^{n-1}\\\\ \dfrac{1}{3}^n\cdot \dfrac{1}{3}^{-1}= \dfrac{1}{2.187}\\\\ \dfrac{1}{3^1}^n\cdot \dfrac{1}{3^1}^{-1}= \dfrac{1}{3^7}\\\\ 3^{-n}\cdot3^{-1\cdot}^{(-1)}=3^{-7}\\ 3^{-n}\cdot3^1=3^{-7}\\\\ 3^{-n}= \dfrac{3^{-7}}{3^1}\\\\ 3^{-n}=3^{-7}\cdot3^{-1}\\\\ \not3^{-n}=\not3^{-8}\\\\ -n=-8\\\\ \huge\boxed{\boxed{n=8~termos}}$

Na segunda, 

$a_9=a_1\cdot q^8\\ a_9=64\cdot\left( \dfrac{1}{2}\right)^8\\ a_9=64\cdot \dfrac{1}{256}\\ a_9= \dfrac{64}{256}\\\\ a_9= \dfrac{64\div64}{256\div64}\\\\\\ \huge\boxed{\boxed{a_9= \dfrac{1}{4}}}$

Tenha ótimos estudos ;D