Question

If the sum of first $n$ natural numbers is one-fifth of the sum of their squares, then $n$ is

• A. $5$
• B. $6$
• C. $7$
• D. $8$

Answer 1

The correct option is C $7$

Sum of the first n natural numbers $=\frac{n(n+1)}{2}$
Sum of the squares of the first n natural numbers
$=\frac{n(n+1)(2n+1)}{6}$
$\therefore \frac{n(n+1)}{2}=\frac{1}{5}\left(\frac{n(n+1)(2n+1)}{6}\right)$
$\Rightarrow 2n+1=15$

$\Rightarrow n=7$