Question

The sum of an infinite terms of a G.P. is $20$ and sum of their squares is $100.$ If $r$ is the common ratio of the G.P., then the value of $10r$ is

Answer 1

Let $a$ be the first term of the G.P.
$a+ar+a{r}^2+\cdots \text{upto}\infty =20$
$\Rightarrow \frac{a}{1-r}=20,|r|<1\cdots \left(1\right)$

Also, $a^2+(ar{)}^2+(a{r}^2{)}^2+\cdots \text{upto}\infty =100$
$\Rightarrow \frac{a^2}{1-r^2}=100$
Using $\left(1\right),$ we get
$\frac{\left[20\right(1-r){]}^2}{1-r^2}=100$
$\Rightarrow 4(1+r^2-2r)=1-r^2$
$\Rightarrow 5r^2-8r+3=0$
$\Rightarrow r=1,\frac{3}{5}$
But $|r|<1.$ Hence, $r=\frac{3}{5}$
And, $10r=10\left(\frac{3}{5}\right)=6$