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Q.
In a data, if the number $i$ is repeated $i$ times for $i=1,2, \ldots, n$, then the mean of the data is
TS EAMCET 2015
Solution:
Since, $i$ is repeated $i$ times for $i=1,2,3, \ldots, n$
Then, mean of given data will be
$\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} $