Sum of 1 to n natural numbers is 36, then find the value of n.

1.	Sum of 1 to n natural numbers is 36, then find the value of n.
Answer» The natural numbers from 1 to n are 1,2, 3, ……, n. The above sequence is an A.P. ∴ a = 1, d = 2 – 1 = 1 S_n = 36 …[Given] Now, S_n = [2a + (n – 1)d] ∴ 36 = n/2 [2(1) + (n – 1)(1)] ∴ 36 = (n/2) (2 + n – 1) ∴ 36 × 2 = n (n + 1) ∴ 72 = n (n + 1) ∴ 72 = n² + n ∴ n² + n – 72 = 0 ∴ n² + 9_n – 8n – 72 = 0 ∴ n(n + 9) – 8 (n + 9) = 0 ∴ (n + 9) (n – 8) = 0 ∴ n + 9 = 0 or n – 8 = 0 ∴ n = -9 or n = 8 But, n cannot be negative. ∴ n = 8 ∴ The value of n is 8.

Answer»

The natural numbers from 1 to n are 1,2, 3, ……, n.

The above sequence is an A.P.

∴ a = 1, d = 2 – 1 = 1

S_n = 36 …[Given]

Now, S_n = [2a + (n – 1)d]

∴ 36 = n/2 [2(1) + (n – 1)(1)]

∴ 36 = (n/2) (2 + n – 1)

∴ 36 × 2 = n (n + 1)

∴ 72 = n (n + 1)

∴ 72 = n² + n

∴ n² + n – 72 = 0

∴ n² + 9_n – 8n – 72 = 0

∴ n(n + 9) – 8 (n + 9) = 0

∴ (n + 9) (n – 8) = 0

∴ n + 9 = 0 or n – 8 = 0

∴ n = -9 or n = 8 But, n cannot be negative.

∴ n = 8

∴ The value of n is 8.

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