Question

Calcule o valor de x e a quantidade de termos da PG ?
PG ( 2x,8x,...,512x),sabendo que a soma dos seus termos é -1364.

Answer 1

$a_{1}=2x\\a_{2}=8x\\\\q=\dfrac{a_{2}}{a_{1}}=\dfrac{8x}{2x}=4$
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Achando a quantidade de termos com a fórmula do termo geral:

$a_{n}=a_{1}\cdot q^{n-1}\\512x=2x\cdot4^{n-1}\\256=4^{n-1}\\4^{4}=4^{n-1}\\n-1=4\\n=4+1\\\\\boxed{\boxed{n=5}}$

Como a soma dos seus cinco termos é -1364:

$S_{5}=-1364\\\\\\\dfrac{a_{1}(q^{5}-1)}{q-1}=-1364\\\\\\\dfrac{2x\cdot(4^{5}-1)}{4-1}=-1364\\\\\\\dfrac{x\cdot(1024-1)}{3}=-682\\\\\\\dfrac{x\cdot1023}{3}=-682\\\\\\x=\dfrac{-682\cdot3}{1023}\\\\\\x=\dfrac{-682}{341}\\\\\\\boxed{\boxed{x=-2}}$