Question

$\frac{\frac{1}{2}.\frac{2}{2}}{1}+\frac{\frac{2}{2}.\frac{3}{2}}{1^3+2^3}+\frac{\frac{3}{2}.\frac{4}{2}}{1^3+2^3+3^3}.....+nterms=$

• A. $\frac{n^2}{{(n+1)}^2}$
• B. $\frac{n^3}{{(n+1)}^3}$
• C. $\frac{n}{n+1}$
• D. $\frac{2n}{{(n+1)}^3}$

Answer 1

Answer: (C)

$\frac{n}{n+1}$

Given series is

$\frac{\frac{1}{2}.\frac{2}{2}}{1}+\frac{\frac{2}{2}.\frac{3}{2}}{1^3+2^3}+\frac{\frac{3}{2}.\frac{4}{2}}{1^3+2^3+3^3}.....+nterms$

Here, the general term is $\frac{\frac{n}{2}\times \frac{n+1}{2}}{{\left[\frac{n(n+1)}{2}\right]}^2}=\frac{1}{n(n+1)}=\frac{1}{n}-\frac{1}{n+1}$

Sum of $n$ terms is $1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...-\frac{1}{n+1}=1-\frac{1}{n+1}=\frac{n}{n+1}$