Find a $G.P.$ for which sum of the first two terms is $-4$ and the fifth term is $4$ times the third term.
Find a $G.P.$ for which sum of the first two terms is $-4$ and the fifth term is $4$ times the third term.
Let $a$ be the first term and $r$ be the common ratio of the $G.P.$
According to the given conditions,
$A_{2}=-4=\frac{a\left(1-r^{2}\right)}{1-r}$ .......$(1)$
$a_{5}=4 \times a_{3}$
$\Rightarrow a r^{4}=4 a r^{2} \Rightarrow r^{2}=4$
$\therefore r=\pm 2$
From $(1),$ we obtain
$-4=\frac{a\left[1-(2)^{2}\right]}{1-2}$ for $r=2$
$\Rightarrow-4=\frac{a(1-4)}{-1}$
$\Rightarrow-4=a(3)$
$\Rightarrow a=\frac{-4}{3}$
Also, $-4=\frac{a\left[1-(-2)^{2}\right]}{1-(-2)}$ for $r=-2$
$\Rightarrow-4=\frac{a(1-4)}{1+2}$
$\Rightarrow-4=\frac{a(-3)}{3}$
$\Rightarrow a=4$
Thus, the required $G.P.$ is $\frac{-4}{3}, \frac{-8}{3}, \frac{-16}{3}, \ldots$ or $4,-8,-16,-32 \ldots$
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