Question

The 19^th term of an AP is equal to 3 times its 6^th term. If its 9^th term is 19, find the AP. 

Answer 1

Let a be the first term and d be the common difference of the AP. Then,

`a_19 = 3a_6`                 (Given)

⇒ a + 18d = 3 (a+ 5d)                     [ a_n  = a + (n-1) d]

⇒ a + 18d = 3a + 15d

⇒ 3a - a = 18d - 15d

⇒ 2a = 3d           ................(1)

Also,

a₉  = 19             (Given)

⇒ a +8d = 19          ..............(2)

From (1) and (2), we get

`(3d)/2 + 8d = 19`

`⇒  (3d +16d)/2 = 19`

⇒ 19d =38

⇒ d =2

Putting d = 2 in (1), we get

2a = 3 × 2=6

⇒ a = 3

So,

a₂ = a + d = 3+2 = 5

a₃ = a +2d = 3+2 ×  2=7
Hence, the AP is 3,5,7,9,........