The mean of n observations 1, 4, 9, 16, \dots, n^{2} is 130. What is the value of n?

Explanation

The mean of the squares of the first n natural numbers is \frac{\sum n^2}{n} = \frac{n(n+1)(2n+1)}{6n} = \frac{(n+1)(2n+1)}{6}. Given that the mean is 130, we have (n+1)(2n+1) = 780. By testing the given options, if n=19, then (19+1)(2(19)+1) = 20 \times 39 = 780. Thus, n=19.

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