Question

Show that the ratio of the sum of first n terms of a G.P. to the sum of terms from $(n+1{)}^{th}to(2n{)}^{th}$ term is $\frac{1}{r^n}$.

Answer 1

Sum of first n term of G.P.

$=a+a_2+a_3+....+a_n$

$=a+ar+a{r}^2+....+a{r}^{n-1}$ $[\because t_n=a{r}^{n-1}].....\left(i\right)$

Also sum of term from

$n+1$th to (2n) th term is

$=a_{n+1}+a_{n+2}+...+a_{2n}$

$=a{r}^n+a{r}^{n-1}+....+a{r}^{2n-1}$ .......(ii)

Ratio of (i) and (ii) is

$=\frac{a+ar+a{r}^2+....a{r}^{n-1}}{a{r}^n+a{r}^{n-1}+....+a{r}^{2n-1}}$

$=\frac{\frac{a(1-r^n)}{1-r}}{\frac{a{r}^n(1-r^n)}{1-r}}$ ([\because S_n = \frac{a(1 - r^n)}{1 - r}])

$=\frac{1}{r^n}$.