Question

If $S_1,S_2,S_3$ are respectively the sum of n, 2n and 3n terms of a G.P. Then $S_1(S_3-S_2)=(S_2-S_1{)}^2$.

• A. True
• B. False

Answer 1

The correct option is A True

$S_1=$Sum of n terms in G.P.

$S_2=$Sum of 2n terms in G.P.

$S_3=$4Sum of 3n terms in G.P.

$S_1=\frac{a(r^n-1)}{r-1},S_2=\frac{a(r^{2n}-1)}{r-1},S_3=\frac{a(r^{3n}-1)}{r-1}$

a is a first term and r is a common ratio.

$S_3-S_2=\frac{a(r^{3n}-r^{2n})}{r-1}{S}_1(S_3-S_2)=\frac{a^2(r^{4n}-2r^{3n}+r^{2n})}{(r-1{)}^2}=\frac{a^2(r^{2n}-r^n{)}^2}{(r-1{)}^2}=\frac{a^2\left(\right(r^{2n}-{1\left)\right(r}^n-1){)}^2}{(r-1{)}^2}={\left[\frac{a(r^{2n}-{1)-a(r}^n-1)}{(r-1)}\right]}^2{S}_1(S_3-S_2)=(S_2-S_1{)}^2$