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

First term of first AP = A = 63 and common difference = D = 65 – 63 = 2
Therefore, A_n = A + (n -1)D ⇒ A_n = 63 + (n -1)2
First term of second AP = a = 3 and common difference = d = 10 – 3 = 7
Therefore, a_n = a + (n -1)d ⇒ a_n = 3 +(n -1)7
According to question, A_n = a_n
⇒ 63 + (n – 1)2 = 3 + (n -1)7 ⇒ 63 + 2_n – 2 = 3 + 7_n – 7
⇒ 65 = 5_n ⇒ n = 13
Hence, the 13th term of both the APs are equal.