If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum of first n terms.

Here, S₇ = 49 and S₁₇ = 289.
The sum of n terms of an AP is given by
S_n = n/2[2a + (n – 1)d]
⇒ S₇ = 7/2[2a +(7 -1) d]
⇒ 49 = 7/2[2a + 6d]
⇒ 49 = 7(a + 3d)
⇒ 7 = a + 3d
⇒ a = 7 – 3d …(1)
and S₁₇ = 17/2[2a + 17 – 1)d)
⇒ 289 = 17/2[2a + 16d]
⇒ 289 = 17(a +8d)
⇒ 17 = a + 8d
Putting the value of a from equation (1), we get
⇒ 17 = 7 – 3d + 8d
⇒ 5d = 10
⇒ d = 2
Putting the value of d in equation (1), we get
⇒ a = 7 – 3 × 2 = 1
The sum of n terms of AP is given by
S_n = n/2[2a + (n – 1)d]
= n/2[2(1) + (n -1)(2)]
= n/2[2 + 2n – 2] = n²