2016/108) If $S_n$ is sum of n terms of a GP show that $S_n(S_{3n} - S_{2n}) = (S_{2n} - S_{n})^2$

we have let 1st term be a and ratio be t
so $S_p = a\frac{t^{p}- t}{t-1}$
we have LHS
$= (a\frac{t^{n}- 1}{t-1})(\frac{a(t^{3n}- 1) -  a(t^{2n}- 1)}{t-1}$
$= \frac{(at^n-1)(a(t^{3n} - t^{2n}}{(t-1)}^2$
$= \frac{a^2t^2(t^n-1)^2}{(t-1)^2}$
$=(\frac{at^n(t^n-1)}{t-1})^2$
RHS = $(S_{2n} - S_{n})^2 = (\frac{at^{2n-1} - 1 - at^n + 1}{t-1})^2 =    (\frac{at^{2n}-at^n}{t-1})^2=LHS$