Question

Sum of infinite number of terms in G.P. is 20 and sum of their square is 100. The common ratio of G.P. is

Answer 1

$\frac{a}{1-r}=20,soa=20(1-r)=20-20r$
The sum of the squares of its terms is:
$a^2+(ar{)}^2+(a{r}^2{)}^2+(a{r}^3{)}^2+...=a^2+a^2{r}^2+a^2{r}^4+a^2{r}^6+...$
This is a new geometric series with first term $a^2$ and common ratio $r^2$, so its sum is:
$a^2/(1-r^2)=100a^2=100(1-r^2)a^2=100-100r^2(20-20r{)}^2=100-100r^{2}400-800r+400r^2=100-100r^{2}4-8r+4r^2=1-r^{2}4-8r+4r^2-1+r^2=05r^2-8r+3=0$
$r=\frac{(-(-8)+\sqrt{8^2}-4\times 5\times 3)}{(2\times 5)}r=\frac{(8+(-\sqrt{64-60}\left)\right)}{10}r=\frac{(8+(-\sqrt{4}\left)\right)}{10}r=\frac{(8+(-2\left)\right)}{10}r=\frac{10or6}{10}r=1or0.6$
But if r = 1 then a = 20(1 - 1) = 20 $\times$0 = 0. That means every term is zero, so it can't have a sum of 20, so that doesn't work. So r $=0.6=\frac{3}{5}$.