Question

If $S_1,S_2,S_3\cdots S_p$ are the sums of infinite geometric series whose first terms are 1,2,3,... p and whose common
ratios are $\frac{1}{2},\frac{1}{3},\frac{1}{4},\cdots \frac{1}{p+1}respectively,$
Then $S_1+S_2+S_3+\cdots +S_p=$

• A. $\frac{p(p+1)}{2}$
• B. $\frac{p(p+3)}{2}$
• C. p
• D. None of these

Answer 1

The correct option is B $\frac{p(p+3)}{2}$

$S_1=\frac{a}{1-r}=\frac{1}{1-\frac{1}{2}}=2,S_2=\frac{2}{1-\frac{1}{3}}=\frac{2}{\frac{2}{3}}=3,S_3=\frac{3}{1-\frac{1}{4}}=\frac{3}{\frac{3}{4}}=4S_P=\frac{p}{1-\frac{1}{p+1}}\frac{p}{\frac{p}{p+1}}=p+1\therefore S_1+S_2+S_3+\cdots +S_p=\frac{p}{2}(2+p+1)=\frac{p(p+3)}{2}$