Question

If the sum of an infinite decreasing G.P. is 3 and the sum of the squares of its term is $\frac{9}{2}$, then write its firstterm and common difference.

Answer 1

Let G.P. be $a,ar,a{r}^2,.......$

$S_{\infty }^{\prime }=\frac{a}{1-r}$

$3=\frac{a}{1-r}$ ....... (i)

Square of terms

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

$S_{\infty }^n=\frac{a^2}{1-r^2}$

$\frac{9}{2}=\frac{a^2}{1-r^2}$ ...... (ii)

Using equation (i) and (ii),

$\frac{9}{2\times 3}=\frac{a^2}{1-r^2}\times \frac{1-r}{a}$

$\frac{3}{2}=\frac{a}{1+r}$ ........ (iii)

Using equation (iii) and (ii),

$\frac{3}{2\times 3}=\frac{a}{1+r}\times \frac{1-r}{a}$

$\frac{1}{2}=\frac{1-r}{1+r}$

$1+r=2-2r$

$3r=1$

$r=\frac{1}{3}$

From equation (i),

$3=\frac{a-r}{1-\frac{1}{3}}$

$a=3\times \frac{2}{3}$

$a=2$

$r=\frac{1}{3}$