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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.
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Solution
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)
\[\Rightarrow a_1 + a_2 + . . . + a_{2n} = 5 \left( a_1 + a_3 + a_5 + . . . + a_{2n - 1} \right)\]
\[ \Rightarrow a + ar + . . . + a r^{2n - 1} = 5 \left( a + a r^2 + . . . + a r^{2n - 2} \right)\]
\[ \Rightarrow a\left( \frac{1 - r^{2n}}{1 - r} \right) = 5a\left\{ \frac{1 - \left( r^2 \right)^n}{1 - r^2} \right\} \]
\[ \Rightarrow 1 + r = 5 \]
\[ \therefore r = 4\]
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