Q52 / 160 · TS EAMCET 2015 · Mathematics

In a data, if the number $i$ is repeated $i$ times for $i=1,2, \ldots, n$, then the mean of the data is

  • A
    $\frac{2 n+1}{6}$
  • B
    $\frac{2 n+1}{4}$
  • Correct answer
    C
    $\frac{2 n+1}{3}$
  • D
    $\frac{2 n+1}{2}$

Solution

Since, $i$ is repeated $i$ times for $i=1,2,3, \ldots, n$. Then, mean of given data will be $$ \begin{aligned} \bar{X} & =\frac{1 \times 1+2 \times 2+3 \times 3+\ldots+n \times n}{1+2+3+\ldots+n}=\frac{\Sigma n^2}{\Sigma n} \\ & =\frac{\frac{n(n+1)(2 n+1)}{6}}{\frac{n(n+1)}{2}}=\frac{2 n+1}{3} \end{aligned} $$

Hence, the correct answer is option: 'C'.

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