Question

If $r>1$ and
$x=a+\frac{a}{r}+\frac{a}{r^2}$
$y=b+\frac{b}{r}+\frac{b}{r^2}$
$z=c+\frac{c}{r}+\frac{c}{r^2},$

then the value of $\frac{xy}{z^2}$ is ___.

• A. $\frac{ab}{c^2}$
• B. $\frac{abr}{c}$
• C. $\frac{ab}{c^{2}r}$
• D. $\frac{ab}{c}$

Answer 1

The correct option is A

$\frac{ab}{c^2}$

Here, we will calculate the value of $x,y$ and $z$ in terms of $a,b,c$ and $r.$
The sum of a GP with first term $a$ and common ratio $r$ is given by
$S=\frac{a}{1-r}.$

$\Rightarrow x=\frac{a}{1-r}$

$y=\frac{b}{1-r}$

$z=\frac{c}{1-r}$

$\Rightarrow \frac{xy}{z^2}=\frac{\frac{a}{(1-r)}\times \frac{b}{(1-r)}}{\frac{c^2}{{(1-r)}^2}}$

$=\frac{ab}{c^2}$