Question

Sum of 1 to n natural number is 45, then find the value of n.

Answer 1

Natural numbers are 1, 2, 3 ......... n in AP.

Where, a = 1, d = 1

S_n = 45

∴ S_n = `n/2 [2a + (n - 1)d]`

⇒ 45 = `n/2 [2 xx 1 + (n - 1) xx 1]`

⇒ 90 = n[2 + n – 1]

∴ 90 = n[n + 1]

∴ n² + n – 90 = 0

⇒ n² + 10n – 9n – 90 = 0

⇒ n(n + 10) – 9(n + 10) = 0

⇒ (n + 10)(n – 9) = 0

∴ n + 10 = 0

or n – 9 = 0

∴ n = – 10

or n = 9

∴ n cannot be negative.

∴ n = 9