The mean of the values of 1,2,3,.......n with respectively frequencies x, 2x, 3x, ..........nx, is

Question

The mean of the values of $1,2,3, \ldots, n$ with respectively frequencies $x, 2 x, 3 x, \ldots, n x$, is

(a) $\frac{n}{2}$

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

(c) $\frac{1}{6}(2 n+1)$

(d) $\frac{n}{2}$

Answer 1

$\Sigma f=x+2 x+3 x+\ldots+n x$

$\begin{aligned}& =x(1+2+3+\ldots+n) \\& =\frac{1}{2} n x(n+1) \\\Sigma(f \times x) & =1 \times x+2 \times 2 x+3 \times 3 x+\ldots+n \times n x \\& =x\left(1^2+2^2+3^2+\ldots+n^2\right)\end{aligned}$

$\begin{aligned}& =x \cdot \frac{1}{6} n(n+1)(2 n+1) \\\text { Mean } & =\frac{\Sigma(f \times d)}{\Sigma f}=\frac{1}{3}(2 n+1)\end{aligned}$

So, The correct option of this question will be (B).