Find the middle term of the AP 6, 13, 20, ..., 216.

First term of the AP = 6

Common difference = d = 13 - 6 = 7

Last term = 216

Since

a_n = a + (n - 1) × d

∴ 216 = 6 + (n - 1) × 7

⇒ 216 - 6 = 7n - 7

⇒ 210 = 7n - 7

⇒ 210 + 7 = 7n

⇒ 7n = 217

⇒ n = 217/7 = 31

∴ Middle term is (31 + 1)/2 = 16^th

So, a₁₆ = a + (16 - 1) × d

∴ a₁₆ = 6 + 15 × 7

⇒ a₁₆ = 6 + 105 = 111

∴ Middle term of the AP is 111.