1. Tardigrade
  2. Question
  3. Mathematics
  4. If the mean of the squares of first n natural numbers be 11, then n is equal to

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. If the mean of the squares of first n natural numbers be 11, then n is equal to

Statistics

Solution:

$ n = \frac{1}{n} \left\{1^{2}+2^{2}+3^{2} + ... +n^{2}\right\}$
$ \Rightarrow 11 = \frac{1}{n} \left\{ \frac{n\left(n+1\right)\left(2n+1\right)}{6}\right\}$
$ \Rightarrow 2n^{2} + 3n + 1 = 66$
$ \Rightarrow 2n^{2} + 3n -65 = 0$
$ \Rightarrow n = \frac{-3 \pm\sqrt{9+520}}{4} $
$\Rightarrow n = \frac{-3 \pm23}{4} = 5, - \frac{13}{2} ;$
but $ n \le0 $ therefore, n = 5.