If the 10th term of an AP is 52 and 17th term is 20 more than its 13th term, find the AP.

Given: 10^th term of the AP is 52.

17^th term is 20 more than the 13^th term.

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

Since,

a_n = a + (n - 1) × d

therefore for 10^th term, we have,

52 = a + (10 - 1) × d

⇒ 52 = a + 9d ………… (1)

Now, 17^th term is 20 more than the 13^th term.

∴ a₁₇ = 20 + a₁₃

⇒ a + (17 - 1)d = 20 + a + (13 - 1)d

⇒ 16d = 20 + 12d

⇒ 4d = 20

⇒ d= 5

∴ from equation (1), we have,

52 = a + 9d

⇒ 52 = a + 9 × 5

⇒ 52 = a + 45

⇒ a = 52 - 45

⇒ a = 7

∴ AP is a, a + d, a + 2d, a + 3d,…

∴ AP is 7, 12, 17, 22….