Question

If $S_1,S_2,S_3,{...S}_n$ are the sums of infinite geometric series whose first terms are $1,2,3,.....,n$ and whose common ration are $\frac{1}{2},\frac{1}{3},\frac{1}{4},...,\frac{1}{n+1}$ respectively, then find the value of
${S_1}^2{{+S}_2}^2{{+S}_3}^2+....+{S_{2n-1}}^2.$

Answer 1

$S_r=r+1$ $\therefore S_r^2={(r+1)}^2$
$\therefore S_1^2+S_2^2+..S_{2n-1}^2=\stackrel{2n-1}{\underset{r=1}{\Sigma }}{s}_r^2$
$=2^2+3^2+4^2+..(2n{)}^2$
$[1^2+2^2+3^2+..N^2]-1$
Where $N=2n$
$\frac{1}{6}N(N+1)(2N+1)-1$
$\frac{1}{3}n(2n+1)(4n+1)-1$