Question

The sum of an infinite number of terms of a G.P. is $20$, and the sum of their squares is $100$, then the first term of the G.P. is

• A. $5$
• B. $\frac{8}{5}$
• C. $\frac{3}{5}$
• D. $8$

Answer 1

The correct option is D $8$

$a+ar+a{r}^2+\cdots \text{to}\infty =20$
$\Rightarrow \frac{a}{1-r}=20...\left(1\right)$
$a^2+a^2{r}^2+a^2{r}^4+\cdots \text{to}\infty =100$
$\Rightarrow \frac{a^2}{1-r^2}=100...\left(2\right)$
Squaring $\left(1\right),$ we have
$\frac{a^2}{(1-r{)}^2}=400...\left(3\right)$

Dividing $\left(3\right)$ by $\left(2\right),$ we get
$\frac{\frac{a^2}{(1-r{)}^2}}{\frac{a^2}{1-r^2}}=\frac{400}{100}$
$\Rightarrow \frac{1-r^2}{(1-r{)}^2}=4$
$\Rightarrow \frac{1+r}{1-r}=4$
$\Rightarrow 1+r=4-4r$
$\Rightarrow 5r=3$
$\Rightarrow r=\frac{3}{5}$

Now using equation $\left(1\right)$,
$\Rightarrow \frac{a}{1-\frac{3}{5}}=20\therefore a=8$