Question

The sum S of first n natural numbers is given by the relation S = $\frac{n(n+1)}{2}$ Find $n$ if the sum is 276

Answer 1

We have S = $\frac{n(n+1)}{2}=276$
$\Rightarrow n^2+n-552=0$
This gives n = $\frac{-1+\sqrt{1+2208}}{2},\frac{-1-\sqrt{1+2208}}{2}$
$So\Rightarrow n=\frac{-1+\sqrt{2209}}{2},\frac{-1-\sqrt{2209}}{2}$
$\Rightarrow n=\frac{-1+47}{2},\frac{-1-47}{2}$
$\Rightarrow$ $n=23,-24$
We reject $n=-24$ since $-24$ is not a natural number
Therefore $n=23$