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

Let 'a' be the first term and 'r' be the common ratio of the given G.P

Then, $\frac{\text{[sum of first n terms]}}{\left[\text{sum of terms from}\right(n+1{)}^{th}\text{to}(2n{)}^{th}}$

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

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