If the sum of first 7 terms of an A.P is 49 and that of its first 17 terms is 289, find the sum of first n terms of the A.P. (CBSE 2013)

Let the first term and common difference of given AP be a and d respectively. Let the sum of first n terms is denoted by S_n.

Given-

Sum of first 7 terms (S₇) = 49

⇒ (7/2) × [2a+(7-1)d] = 49

[∵ S_n = (n/2)[2a+(n-1)d]

⇒ 2a+6d = 14

⇒ a+3d =7…(1)

and, Sum of first 17 terms (S₁₇) = 49

⇒ (17/2) × [2a+(17-1)d] = 289

[∵ S_n = (n/2)[2a+(n-1)d]

⇒ 2a+16d = 34

⇒ a+8d =17…(2)

Subtracting equation (1) from equation (2), we get-

5d = 10

∴ d = 2

Substituting the value of d in equation (2), we get-

a = 1

Thus, Sum of first n terms of AP(S_n) = (n/2)[2a+(n-1)d

= (n/2)[2+(n-1)2)]

= (n/2)(2n)

= n²