Question

If the sum of an infinite $GP$ is $20$ and sum of their square is $100$ then common ration will be=

• A. $1/2$
• B. $1/4$
• C. $3/5$
• D. $1$

Answer 1

The correct option is C $3/5$

Let $G.P$ is $a,ar,a{r}^2,....\infty$

sum $=\frac{a}{1-r}=20---\left(1\right)$

If terms are required the $G.P.$

becomes : $a^2,a^2{r}^2,a^2{r}^4,.....\infty$

Sum $=\frac{a^2}{1-r^2}=100---\left(2\right)$

Name dividing square of equation $\left(1\right)$ by equation $\left(2\right)$

$\frac{\frac{a^2}{1-r^2}}{\frac{a^2}{(1-r{)}^2}}=\frac{100}{400}$

$\Rightarrow \frac{(1-r)}{(1-r)(1+r)}=\frac{1}{4}$

$\Rightarrow \frac{1-r}{1+r}=\frac{1}{4}$

$\Rightarrow 4-4r=1+r$

$\Rightarrow 4-1=4r+r$

$\Rightarrow 5r=3$

$\Rightarrow r=\frac{3}{5}$