Question

The simple and weighed arithmetic means of the first n natural numbers, the weights being the corresponding numbers are respectively.

• A. $\frac{n+1}{2},\frac{2n+1}{2}$
• B. $\frac{n+1}{2},\frac{2n+1}{4}$
• C. $\frac{n+1}{2},\frac{2n+1}{3}$
• D. $n,n^2$

Answer 1

The correct option is C $\frac{n+1}{2},\frac{2n+1}{3}$

First $n$ natural no.$=1,2,3,....,n$

Mean$=\overline{x}=\frac{1+2+3+.....+n}{n}$

Weighted mean $=\frac{1.1+2.2+3.3+.....+n.n}{1+2+3+...+n}=\frac{\sum n^2}{\sum n}$

Weighted mean$=\frac{\frac{n(n+1)(2n+1)}{6}}{\frac{n(n+)}{2}}=\frac{2n+1}{3}$