Question

The 9^th term of an A.P. is equal to 6 times its second term. If its 5^th term is 22, 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,

a₉ = 6a₂
⇒ a + (9 − 1)d =  6(a + (2 − 1)d)
⇒ a + 8d =  6a + 6d
⇒ 8d − 6d =  6a − a
⇒ 2d = 5a
⇒ a = \[\frac{2}{5}\]d   .... (1)
Also, a₅ = 22
⇒ a + (5 − 1)d = 22
⇒ a + 4d = 22   ....(2)
On substituting the values of (1) in (2), we get \[\frac{2}{5}\]d + 4d = 22
⇒ 2d + 20d = 22 × 5
⇒ 22d = 110
⇒ d = 5
⇒ a =\[\frac{2}{5} \times 5\]    [From (1)]
⇒ a = 2

Thus, the A.P. is 2, 7, 12, 17, .... .