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

Question

Philoid · Accepted Answer

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

Given: a₉ = 19

a₁₉ = 3 a₆

Now, Consider a₉ = 19

⇒ a + (9 - 1)d = 19

⇒ a + 8d = 19 ………………….(1)

Consider a₁₉ = 3 a₆

⇒ a + 18d = 3(a + 5d)

⇒ a + 18d = 3a + 15d

⇒ 2a - 3d = 0 ………………….(2)

Now, subtracting twice of equation (1) from (2), we get,

- 19d = - 38

⇒ d = 2

∴ from equation (1), we get,

a = 19 - 8d

⇒ a = 19 - 8 × 2

⇒ a = 19 - 16

⇒ a = 3

Thus the AP is a, a + d, a + 2d, a + 3d, a + 4d,….

Therefore the AP is 3, 5, 7, 9….