For what value of n, are the nth terms of two APs: 63, 65, 67, . . . and 3, 10, 17, . . . equal?

In first AP: a = 63, d = 2

In second AP: a = 3, d = 7

As per question:

63 + (n – 1) 2 = 3 + (n – 1) 7

⇒ 63 – 3 + (n – 1) 2 = (n – 1) 7

⇒ 60 + 2n – 2 = 7n – 7

⇒ 2n + 58 = 7n – 7

⇒ 2n + 58 + 7 = 7n

⇒ 2n + 65 = 7n

⇒ 7n – 2n = 65

⇒ 5n = 65

⇒ n = 65/5 = 13

Thus, for the 13 value of n, nth term of given two APs will be equal