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
$1+2+3+...+n=\frac{1}{5}[1^2+2^2+3^2+...+n^2]$
$\Rightarrow$ $\Sigma n=\frac{1}{5}\Sigma n^2$
$\Rightarrow$ $\frac{n(n+1)}{2}=\frac{1}{5}\left[\frac{n(n+1)(2n+1)}{6}\right]$
$\Rightarrow$ $\frac{1}{2}=\frac{(2n+1)}{30}$
$\Rightarrow$ $2n+1=15$
$\Rightarrow$ $2n=14$
$\Rightarrow$ $n=7$