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In a data, if the number $i$ is repeated $i$ times for $i=1,2, \ldots, n$, then the mean of the data is
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Verified Answer
The correct answer is:
$\frac{2 n+1}{3}$
Since, $i$ is repeated $i$ times for $i=1,2,3, \ldots, n$. Then, mean of given data will be
$$
\begin{aligned}
\bar{X} & =\frac{1 \times 1+2 \times 2+3 \times 3+\ldots+n \times n}{1+2+3+\ldots+n}=\frac{\Sigma n^2}{\Sigma n} \\
& =\frac{\frac{n(n+1)(2 n+1)}{6}}{\frac{n(n+1)}{2}}=\frac{2 n+1}{3}
\end{aligned}
$$
$$
\begin{aligned}
\bar{X} & =\frac{1 \times 1+2 \times 2+3 \times 3+\ldots+n \times n}{1+2+3+\ldots+n}=\frac{\Sigma n^2}{\Sigma n} \\
& =\frac{\frac{n(n+1)(2 n+1)}{6}}{\frac{n(n+1)}{2}}=\frac{2 n+1}{3}
\end{aligned}
$$
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