For what value of n, the nth term of A.P. 63, 65, 67, ….. and nth term of A.P. 3, 10, 17, …… are equal ?

By formula,

a_n = a + (n - 1)d

For first series :

63, 65, 67, …..

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

So,

a_n = 63 + 2(n - 1)

= 63 + 2n - 2

= 61 + 2n.

For second series :

3, 10, 17, …..

First term (a) = 3 and common difference (d) = 10 - 3 = 7.

So,

a_n = 3 + 7(n - 1)

= 3 + 7n - 7

= 7n - 4.

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

∴ 61 + 2n = 7n - 4

⇒ 7n - 2n = 61 + 4

⇒ 5n = 65

⇒ n = $\dfrac{65}{5}$ = 13.

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