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

Taking first AP, 63, 65, 67, … First term, a = 63 Common difference, d = a2−a1 = 65−63 = 2 We know, nth term of this A.P. = an = a+(n−1)d an= 63+(n−1)2 = 63+2n−2 an = 61+2n ………………………………………. (i) Taking second AP, 3, 10, 17, … First term, a = 3 Common difference, d = a2 − a1 = 10 − 3 = 7 We know that, nth term of this A.P. = 3+(n−1)7 an = 3+7n−7 an = 7n−4 ……………………………………………………….. (ii) Given, nth term of these A.P.s are equal to each other. Equating both these equations, we get, 61+2n = 7n−4 61+4 = 5n 5n = 65 n = 13 Therefore, 13th terms of both these A.P.s are equal to each other.