Question

If the sum of first n natural number is $ \frac{1}{5} $ times the sum of their squares, then the value of n is

• A. 5
• B. 6
• C. 7
• D. 8

Answer 1

Answer: (C)

Let first n natural numbers are 1, 2, 3, 4, ..., n, then
$ 1+2+3+...+n=\frac{1}{5}[{{1}^{2}}+{{2}^{2}}+{{3}^{2}}+...+{{n}^{2}}] $
$ \Rightarrow $ $ \Sigma n=\frac{1}{5}\Sigma {{n}^{2}} $
$ \Rightarrow $ $ \frac{n(n+1)}{2}=\frac{1}{5}\left[ \frac{n(n+1)(2n+1)}{6} \right] $
$ \Rightarrow $ $ \frac{1}{2}=\frac{(2n+1)}{30} $
$ \Rightarrow $ $ 2n+1=15 $
$ \Rightarrow $ $ 2n=14 $
$ \Rightarrow $ $ n=7 $