Question

The sum of 5^th and 9^th terms of an A.P. is 30. If its 25^th term is three times its 8^th term, find the A.P.

Answer 1

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

We know that, n^th term = a_n = a + (n − 1)d

According to the question,

a₅ + a₉ = 30
⇒ a + (5 − 1)d + a + (9 − 1)d = 30
⇒ a + 4d + a + 8d = 30
⇒ 2a + 12d = 30
⇒ a + 6d = 15       .... (1)

Also, a₂₅ = 3(a₈)
⇒ a + (25 − 1)d = 3[a + (8 − 1)d]
⇒ a + 24d = 3a + 21d
⇒ 3a − a = 24d − 21d
⇒ 2a = 3d
⇒ a = \[\frac{3}{2}\] d   ....(2)
Substituting the value of (2) in (1), we get

\[\frac{3}{2}\] d + 6d = 15
⇒ 3d + 12d = 15 × 2
⇒ 15d = 30
⇒ d = 2
⇒ a =  \[\frac{3}{2}\]    [From (1)]
⇒ a = 3
Thus, the A.P. is 3, 5, 7, 9, .... .