For what value of n, the n^th term of A.P. 63, 65, 67, ……. and n^th term of A.P. 3, 10, 17, ….., are equal to each other?

In the A.P. 63, 65, 67, ……. first term = a = 63 and common difference = d = 65 - 63 = 2.

In the A.P. 3, 10, 17, ……. first term = a₁ = 3 and common difference = d₁ = 10 - 3 = 7.

Given, n^th term of both A.P. are equal,

∴ a + (n - 1)d = a₁ + (n - 1)d₁

⇒ 63 + (n - 1)2 = 3 + (n - 1)7

⇒ 63 + 2n - 2 = 3 + 7n - 7

⇒ 61 + 2n = 7n - 4

⇒ 7n - 2n = 61 + 4

⇒ 5n = 65

⇒ n = 13.

Hence, n = 13.