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

Given,

1st AP : 63, 65, 67, …….

For 1st A.P.,

First term (a) = 63 and Common difference (d) = 65 - 63 = 2.

nth term of 1st A.P. :

= a + (n - 1)d

= 63 + 2(n - 1)

= 63 + 2n - 2

= 2n + 61.

2nd A.P. : 3, 10, 17, ………

First term (a₁) = 3 and Common difference (d₁) = 10 - 3 = 7.

nth term of 2nd A.P. :

= a₁ + (n - 1)d₁

= 3 + 7(n - 1)

= 3 + 7n - 7

= 7n - 4.

Since, nth term of both A.P.s are equal.

∴ 2n + 61 = 7n - 4

⇒ 7n - 2n = 61 + 4

⇒ 5n = 65

⇒ n = $\dfrac{65}{5}$ = 13.

Hence, 13th term of both the A.P.'s are equal.