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Q.
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
Sequences and Series
Solution:
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 $\frac{a\left(r^{2n}-1\right)}{r-1}=5 \frac{a\left(r^{2n}-1\right)}{r^{2}-1}$
$\Rightarrow 1=\frac{5}{r+1}$
$\Rightarrow r+1=5 \therefore r=4$