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 G.P.
Sum of the first n terms of the series = $a_1+a_2+a_3+...+a_n$
Similarly, $\mathrm{sum}\mathrm{of}\mathrm{the}\mathrm{terms}\mathrm{from}{\left(n+1\right)}^{\mathrm{th}}\mathrm{to}2n^{\mathrm{th}}\mathrm{term}=a_{n+1}+a_{n+2}+...+a_{2n}$
$$\begin{gathered} \therefore \mathrm{Required}\mathrm{ratio}=\frac{a_1+a_2+a_3+...+a_n}{a_{n+1}+a_{n+2}+...+a_{2n}} \\ =\frac{a+ar+...+a{r}^{n-1}}{a{r}^n+a{r}^{n+1}+...+a{r}^{2n-1}} \\ =\frac{a\left({\displaystyle \frac{1-r^n}{1-r}}\right)}{a{r}^n\left({\displaystyle \frac{1-r^n}{1-r}}\right)} \\ =\frac{1}{r^n} \end{gathered}$$