The sum ‘S’ of the first ‘n’ natural numbers is given by S = n (n + 1)/2 . Find ‘n’, if the sum (S) is 276.

11. The sum ‘S’ of the first ‘n’ natural numbers is given by S = ^{n (n + 1)}/₂ . Find ‘n’, if the sum (S) is 276.

Sol. S = ^{n (n + 1)}/₂

∴ 276 = ^{n (n + 1)}/₂

∴ 276 × 2 = n(n+1)

∴ 552 = n² + n

∴  n² + n – 552 = 0

∴  n² + 24n – 23n – 552 = 0

∴  n (n + 24) (n – 23) = 0

∴  n + 24 = 0 or n – 23 = 0

∴  n = – 24 or n = 23

∵ Number of terms cannot be negative,

∴ n = 23