Question

A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying the odd places. Find the common ratio of the G.P.

Answer 1

Let there be 2n terms in the given G.P. with the first term being a and the common ratio being r.
According to the question
Sum of all the terms = 5 (Sum of the terms occupying the odd places)

\[\Rightarrow a_1 + a_2 + . . . + a_{2n} = 5 \left( a_1 + a_3 + a_5 + . . . + a_{2n - 1} \right)\]

\[ \Rightarrow a + ar + . . . + a r^{2n - 1} = 5 \left( a + a r^2 + . . . + a r^{2n - 2} \right)\]

\[ \Rightarrow a\left( \frac{1 - r^{2n}}{1 - r} \right) = 5a\left\{ \frac{1 - \left( r^2 \right)^n}{1 - r^2} \right\} \]

\[ \Rightarrow 1 + r = 5 \]

\[ \therefore r = 4\]