Question

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

Answer 1

We know that sum of first n natural numbers = $\frac{n(n+1)}{2}$
Therefore by the given condition, we have:
$\frac{n(n+1)}{2}=276$
$\Rightarrow$ n² + n = 552
$\Rightarrow$ n² + n – 552 = 0
On splitting the middle term n as 24n – 23n, we get:
n² + 24n – 23n – 552 = 0
$\Rightarrow$ n(n + 24) – 23(n + 24) = 0
$\Rightarrow$ (n + 24) (n – 23) = 0
$\Rightarrow$ n + 24 = 0 or n – 23 = 0
$\Rightarrow$ n = –24 or n = 23
Since n is a natural number, it cannot be negative.
Thus, n = 23.