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, 

\[\text { sum of the terms from } \left( n + 1 \right)^{th}\text {  to } 2 n^{th} \text { term } = a_{n + 1} + a_{n + 2} + . . . + a_{2n}\]

\[\therefore \text { Required ratio } = \frac{a_1 + a_2 + a_3 + . . . + a_n}{a_{n + 1} + a_{n + 2} + . . . + a_{2n}} \]

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

\[ = \frac{a\left( \frac{1 - r^n}{1 - r} \right)}{a r^n \left( \frac{1 - r^n}{1 - r} \right)} \]

\[ = \frac{1}{r^n}\]