Question

In the series $1+3+6+10+......,$ find the sum to $n$ terms is:

• A. $\frac{n(n+1)(n+2)}{6}$
• B. $\frac{n(n+1)(n+3)}{6}$
• C. $\frac{n(2n+1)(2n+3)}{6}$
• D. none of these

Answer 1

The correct option is A $\frac{n(n+1)(n+2)}{6}$

$1+3+6+10+....$

$={\sum }_1^n\frac{n(n+1)}{2}$

$={\sum }_1^n\frac{n^2}{2}+{\sum }_1^n\frac{n}{2}$

$=\frac{n(n+1)(2n+1)}{2\times 6}+\frac{n(n+1)}{4}$

$=\frac{n(n+1)}{4}\left(\frac{2n+4}{3}\right)=\frac{n(n+1)(n+2)}{6}$