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 ratios are $\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{(n+1)}$ respectively, then find the value of $S_1^2+S_2^2+S_3^2+....+S_6^2$

• A. 140
• B. 139
• C. 91
• D. 140

Answer 1

The correct option is B

139

$S_1=\frac{1}{1-\frac{1}{2}}=2$

$S_2=\frac{2}{1-\frac{1}{3}}=3$

Thus, the respective sums are 2,3,4,5,6,7

Sums of square of these numbers $=\left[\frac{n(n+1)(2n+1)}{6}\right]-1=139.$