If the 8^th term of an A.P. is 37 and the 15^th term is 15 more than the 12^th term, find the A.P.

Also, find the sum of first 20 terms of this A.P.

Given,

⇒ a₈ = a + (8 - 1)d

∴ a + 7d = 37 ……..(i)

Also,

⇒ a₁₅ = a₁₂ + 15

⇒ a + (15 - 1)d = a + (12 - 1)d + 15

⇒ a + 14d = a + 11d + 15

⇒ a - a + 14d - 11d = 15

⇒ 3d = 15

⇒ d = 5.

Substituting value of d in (i) we get,

⇒ a + 7(5) = 37

⇒ a + 35 = 37

⇒ a = 2.

A.P. = a, (a + d), (a + 2d), ……….

= 2, 7, 12, ………..

$S = \dfrac{n}{2}[2a + (n - 1)d] \\[1em] = \dfrac{20}{2}[2 \times 2 + (20 - 1) \times 5] \\[1em] = 10(4 + 19 \times 5) \\[1em] = 10 \times 99 \\[1em] = 990.$

Hence, A.P. = 2, 7, 12, ……….. and sum = 990.