Let n be an odd natural number such that the variance of 1 , 2 , 3 , 4 , … , n is 14 . Then n is equal to ________.

Question

TestHub · Accepted Answer

Variance of n natural number is given by,

Variance $= \frac{\sum n^{2}}{n} - {(\frac{\sum n}{n})}^{2} = \frac{n (n + 1) (2 n + 1)}{6 n} - {(\frac{n (n + 1)}{2 n})}^{2}$

$= (n + 1) [\frac{2 n + 1}{6} - \frac{(n + 1)}{4}] = \frac{(n + 1) (n - 1)}{12} = \frac{n^{2} - 1}{12}$

And it is given that Variance $= 14$

$\Rightarrow \frac{n^{2} - 1}{12} = 14$

$\Rightarrow n^{2} - 1 = 168$

$\Rightarrow n = 13$

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