Question:
For what values of $x$, the numbers $\frac{2}{7}+x,-\frac{7}{2}$ are in G.P?
Solution:
The given numbers are $\frac{-2}{7}, x, \frac{-7}{2}$.
Common ratio $=\frac{x}{\frac{-2}{7}}=\frac{-7 x}{2}$
Also, common ratio $=\frac{\frac{-7}{2}}{x}=\frac{-7}{2 x}$
$\therefore \frac{-7 x}{2}=\frac{-7}{2 x}$
$\Rightarrow x^{2}=\frac{-2 \times 7}{-2 \times 7}=1$
$\Rightarrow x=\sqrt{1}$
$\Rightarrow x=\pm 1$
Thus, for $x=\pm 1$, the given numbers will be in G.P.