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 odd places, the common ratio will be

• A. 2
• B. 5
• C. 3
• D. 4

Answer 1

The correct option is D 4

Let the total number of terms in G.P be $2n$ with first term $A$ and common ratio $R$

then $S_{2n}=A\frac{R^{2n}-1}{R-1}$

and sum of numbers occupying the odd places is:

$S_n=A\frac{R^{2n}-1}{R^2-1}$

in this case, the first term which is at the odd place will remain A but the common ratio will become $R^2$ because the next

term of the sequence will be at third position i.e $A{R}^2$

$So,S_{2n}=5S_n$

$\Rightarrow A\frac{R^{2n}-1}{R-1}=5A\frac{R^{2n}-1}{R^2-1}$ [Sum of GP $=\frac{a(r^n-1)}{r-1}$ when $r>1$]

$\Rightarrow \frac{1}{R-1}=\frac{5}{R^2-1}$

$\Rightarrow \frac{R^2-1}{R-1}=5$

$\Rightarrow R+1=5$

$\Rightarrow R=4$

Hence the correct option is D