Question

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

• A. $\frac{1}{2}$
• B. $\frac{1}{4}$
• C. $\frac{3}{5}$
• D. $1$

Answer 1

Answer: (C)

$\frac{3}{5}$

Let $a,ar,a{r}^2,...$ to $\infty$

Now, sum of infinite G.P$=20$

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

where each term of the above G.P is squared, then the progression becomes

$a^2,a^2{r}^2,a^2{r}^4,.....$

Now, first term $A=a^2$ and common ratio$R=r^2$

Sum of above G.P$=100$

$\Rightarrow \frac{A}{1-R}=100$

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

Squaring $\left(1\right)$ we get

$\frac{a^2}{{(1-r)}^2}=400$ ....$\left(3\right)$

Dividing eqn$\left(2\right)$ by $\left(3\right)$ we get

$\frac{{(1-r)}^2}{1-r^2}=\frac{400}{100}=4$

$\Rightarrow \frac{{(1-r)}^2}{(1-r)(1+r)}=4$

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

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

$\Rightarrow 5r=3$

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