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^2_1 + S^2_2 + S^2_3 +...+ S^2_{2n-1}$.

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^2_1 + S^2_2 + S^2_3 +...+ S^2_{2n-1}$
$ = 2^2 + 3^2 + 4^2 + ... + (2n)^2 $
$ =\frac{1}{6}(2n) (2n+1) (4n+1) - 1 $