Find the weighted arithmetic mean of the first 'n' natural numbers, the weights being the corresponding numbers.

A. $$\frac{{\left\{ {n\left( {n + 1} \right)\left( {2n + 1} \right)} \right\}}}{6}$$

B. $$\frac{{\left\{ {n\left( {n + 1} \right)} \right\}}}{2}$$

C. $$\frac{{\left\{ {\left( {2n + 1} \right)} \right\}}}{3}$$

D. $$n$$

Answer: Option C

Solution(By Examveda Team)

$$\eqalign{ & {\text{Weighted arithmetic mean}} \cr & = \frac{{1 \times 1 + 2 \times 2 + 3 \times 3 + \,.....\,n \times n}}{{1 + 2 + 3 + \,.....\, + n}} \cr & = \frac{{n\left( {n + 1} \right)\left( {2n + 1} \right) \times 2}}{{6 \times n\left( {n + 1} \right)}} \cr & = \frac{{2n + 1}}{3} \cr} $$