Question

If S denotes the sum of an infinite G.P. $S_1$ denotes the sum of the squares of its terms, then prove that the first term and common ratio are respectively $\frac{2S{S}_1}{S^2+S_1}$ and $\frac{S^2-S_1}{S^2+S_1}$.

Answer 1

$S=a+ar+a{r}^2+a{r}^3+....$

$S=\frac{a}{1-r}$ .......(1)

$S_1=a^2+a^2{r}^2+a^2{r}^4+a^2{r}^6.....$

$S_1=\frac{a^2}{1-r^2}$ ........(2)

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

$S^2=\frac{S_1(1-r^2)}{1-r^2}$

$(1-r)S^2=S_1(1+r)$

$S^2-S^{2}r=S_1+S_{1}r$

$S_{1}r+S^{2}r=S^2-S_1$

$r=\frac{S^2-S_1}{S_1+S^2}$

Put r in equation (1)

$S(1-r)=a$

$a=S[1-\frac{S^2-S_1}{S^2+S_1}]$

$a=S\left[\frac{S^2+S_1-S^2+S_1}{S^2+S_1}\right]$

$a=\frac{2S{S}_1}{S^2+S_1}$