Question

If $x=a+\frac{a}{r}+\frac{a}{r^2}+\cdots \infty ,$ $y=b-\frac{b}{r}+\frac{b}{r^2}-\cdots \infty$ and $z=c+\frac{c}{r^2}+\frac{c}{r^4}+\cdots \infty$ for $|r|>1,$ then the value of $\frac{xy}{z}$ is

• A. $\frac{ab}{c}$
• B. $\frac{a{b}^2}{c}$
• C. $\frac{ac}{b}$
• D. $\frac{a^{2}b}{c}$

Answer 1

The correct option is A $\frac{ab}{c}$

We have,
$x=\frac{a}{1-\frac{1}{r}}=\frac{ar}{r-1}y=\frac{b}{1-\frac{-1}{r}}=\frac{br}{r+1}z=\frac{c}{1-\frac{1}{r^2}}=\frac{c{r}^2}{r^2-1}$

Now, $xy=\left(\frac{ar}{r-1}\right)\left(\frac{br}{r+1}\right)$
$\Rightarrow xy=\frac{ab{r}^2}{r^2-1}\therefore \frac{xy}{z}=\frac{ab}{c}$