The sum of the first 7 terms of an AP is 49 and the sum of its first 17 terms is 289. Find the sum of its first n terms.

Let a be the first term and d be the common difference.

Given: S₇ = 49, S₁₇ = 289

To find: sum of first n terms.

Now, consider S₇ = 49

⇒ (7/2)[2a + (7 - 1)d] = 49

⇒ (7/2)[2a + 6d] = 49

⇒ [a + 3d] = 7 …………(1)

Now, consider S₁₇ = 289

⇒ (17/2)[2a + (17 - 1)d] = 289

⇒ (17/2) × [2a + 16d] = 289

⇒ [a + 8d] = 17 …………..(2)

Now, on subtracting equation (2) from equation (1), we get,

5d = 10

⇒ d = 2

∴ from equation (1), we get

a = (7 - 3d)

⇒ a = 7 - 6

⇒ a = 1

∴ a = 1, d = 2

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

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

= (n/2)[2n]

= n²

∴ S_n = n²