The sum of the 2^nd term and the 7^th term of an A.P. is 30. If its 15^th term is 1 less than twice of its 8^th term, find the A.P.

Let first term be a and common difference be d.

According to question,

⇒ a₂ + a₇ = 30

⇒ a + (2 - 1)d + a + (7 - 1)d = 30

⇒ a + d + a + 6d = 30

⇒ 2a + 7d = 30 ………(i)

Also,

⇒ 2a₈ - 1 = a₁₅

⇒ 2[a + (8 - 1)d] - 1 = a + (15 - 1)d

⇒ 2[a + 7d] - 1 = a + 14d

⇒ 2a + 14d - 1 = a + 14d

⇒ 2a - a - 1 = 14d - 14d

⇒ a - 1 = 0

⇒ a = 1.

Substituting value of a in (i) we get,

⇒ 2(1) + 7d = 30

⇒ 2 + 7d = 30

⇒ 7d = 28

⇒ d = 4.

A.P. = a, (a + d), (a + 2d),……….

= 1, (1 + 4), (1 + 2.4),………

= 1, 5, 9,…….

Hence, A.P. = 1, 5, 9,……..