Question

If $S_1,S_2,S_3$ be respectively the sums of $n,2n,3n$ terms of a $G.P.$, then prove that

${S_1}^2+{S_2}^2=S_1(S_2+S_3)$

Answer 1

Subtracting $2S_1{S}_2$ from both sides of given relation, we get
${(S_2-S_1)}^2=S_1(S_2+S_3-{2S}_2)$
or ${(S_2-S_1)}^2=S_1(S_3-S_2)$
This we have proved in part $\left(i\right)$.
Another form:
$a=S_1,b=S_2-S_1,c=S_3-S_2$ and we have proved above that $S_1(S_3-S_2)=(S_2-S_1{)}^2$ i.e., $ac=b^2$
$\therefore a,b,c$ are in $G.P.$