Question

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

• A. 5
• B. 1
• C. 4
• D. 3

Answer 1

Answer: (C)

4

Answer 2

Let the G.P. be $a, ar, ar^2, ..........$ $S = a + ar + ar^2 + ..........+$ to $2n$ term $=\frac{a\left(r^{2n}-1\right)}{r-1}$ The series formed by taking term occupying odd places is $S_{1}=a+ar^{2}+ar^{4}+........$ to $n$ terms $S_{1}=\frac{a\left[\left(r^{2}\right)^{n}-1\right]}{r^{2}-1} \Rightarrow S_{1}=\frac{a\left(r^{2n}-1\right)}{r^{2}-1}$ Now, $S=5S_{1}$ or $\frac{a\left(r^{2n}-1\right)}{r-1}=5 \frac{a\left(r^{2n}-1\right)}{r^{2}-1}$ $\Rightarrow 1=\frac{5}{r+1}$ $\Rightarrow r+1=5 \therefore r=4$