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in Arithmetic Progression by (46.9k points)
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Sum of 1 to n natural numbers is 36, then find the value of n.

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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 

Sn = 36 …[Given] 

Now, Sn = [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 + 9n – 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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