Question

If $S_1,S_2,S_3,...,S_n$ are the sums of infinite geometric series, whose first terms are $1,2,3,...,$ and whose common ratios are $\frac{1}{2},\frac{1}{3},\frac{1}{4},...,\frac{1}{n+1}$ respectively, then find the values of $S_1^2+S_2^2+S_3^2+...+S_{2n-1}^2$.

Answer 1

Consider an infinite GP with first term $1,2,3,..,n$ and common ratios $\frac{1}{2},\frac{1}{3},\frac{1}{4},...,\frac{1}{n+1}$
$\therefore S_1=\frac{1}{1-1/2}=2$
$S_2=\frac{2}{1-1/3}=3$
$:::$
$S_{2n-1}=\frac{2n-1}{1-1/2n}=2n$
$\therefore S_1^2+S_2^2+S_3^2+...+S_{2n-1}^2$
$=2^2+3^2+4^2+...+(2n{)}^2$
$=\frac{1}{6}(2n)(2n+1)(4n+1)-1$