Question

If $x_1,x_2,x_3,....$ are n values of the variable x, then mathematical averages, Arithmetic Mean (A.M.), Geometric Mean (G.M) and Harmonic Mean (H.M) are count by the following formulas's.

$A.M.=\frac{x_1+x_2+x_3+...+x_n}{n}=\frac{1}{n}\left\{\sum _{i=1}^n{x}_1\right\}\nless x_1>0$

$G.M.={(x_1\cdot x_2...x_n)}^{1/n}$ if each $x_1(i=1,2,....,n)$ is positive and

$H.M=\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}}=\frac{1}{\frac{1}{n}{\sum }_{i=1}^n\left(\frac{1}{x_1}\right)}$

* In case of frequency distribution $x/f_1(i=1,2,...,n)$ where $f_1$ is the frequency of the variable $x_r$ then the calculation of A.M.is counted as $A.M=\sum _{i=1}^n{f}_1{x}_1/NwhereN=f_1+f_2....+f_n$

** If $w_1,w_2,....,w_{{}_n}$ be the weight assigned to the n values

$x_1,x_2,....,x_{{}_n}$ then weighted A.M is counted by $\frac{{\sum }_{i=1}^n{w}_1{x}_1}{{\sum }_{i=1}^n{w}_1}$

*** if $G_1,G_2$ are the G. M's of two series of sizes $n_1$ & $n_2$ respectively, then the geometric mean (G M.) of the combined series is counted by $\log (G.M)=\frac{n_1\log G_1+n_2\log G_2}{n_1+n_2}$

On the basis of above information answer the following questions,The weighted A.M of first n natural numbers whose weights are squares the corresponding numbers, is equal to

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

Answer 1

The correct option is D $\frac{3}{2}\frac{n(n+1)}{2n+1}$

Given first n natural numbers i.e., $x_i=i$ (i$=$1, 2, ..., n) and their weight $w_i=i^2$

$\therefore$ $x_1,x_2,x_3,...,x_n=1,2,3,4,...,n$ and their weights

$w_1,w_2,w_3,...,w_n=1^1,2^2,3^2,...,n^2$

$A.M.=\overline{x}=\frac{x_1{w}_1+x_2{w}_2+x_3{w}_3+...+x_n{w}_n}{w_1+w_2+w_3+...+w_n}$

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

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

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

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