Question

For what values of $x$, the numbers $-\frac{2}{7},x,-\frac{7}{2}$ are in $G.P$ ?

Answer 1

Given: $-\frac{2}{7},x,-\frac{7}{2}$ are in G.P.
So, common ratio remains same.
Now, common ratio $=\frac{\text{second term}}{\text{first term}}$
$\Rightarrow$ common ratio $=\frac{x}{(-\frac{2}{7})}=-\frac{7x}{2}\cdots \left(i\right)$
Also, Common ratio $=\frac{\text{Third term}}{\text{Second term}}=\frac{-\frac{7}{2}}{x}=-\frac{7}{2x}\cdots \left(ii\right)$
From $\left(i\right)$ & $\left(ii\right),$ we get
$-\frac{7x}{2}=-\frac{7}{2x}$
$\Rightarrow x^2=1$
$\therefore x=1$ or $-1$
For $x=1,G.P.$ is
$-\frac{2}{7},1,-\frac{7}{2}$ and
For $x=-1,G.P.$ is
$-\frac{2}{7},-1,-\frac{7}{2}$