Question

The sum $S$ of first $n$ natural numbers is given by the formula $S=\frac{n(n+1)}{2}$.If $S=231,$ find $n$

Answer 1

$S=\frac{n(n+1)}{2}$

$\Rightarrow 231=\frac{n(n+1)}{2}$

$\Rightarrow n^2+n-462=0$

$\Rightarrow n^2+22n-21n-462=0$

$\Rightarrow n(n+22)-21(n+22)=0$

$\Rightarrow (n+22)(n-21)=0$

$\Rightarrow n=-22$ or $n=21$

$\therefore n=21$ as $n=-22$ is not a natural number.