Question

For what value of n, are the n^th terms of two APs: \(63, 65, 67, …..\)  and  \(3, 10, 17, …..\) equal?

Answer 1

For AP: \(63, 65, 67, ….\)
\(a = 63\) and \(d = a_2 − a_1 = 65 − 63 = 2\)
n^th term of this A.P.
\(a_n = a + (n − 1) d \)
\(a_n= 63 + (n − 1) 2 = 63 + 2n − 2 \)
\(a_n = 61 + 2n\ \ \ \  ……(1)\)
For AP: \(3, 10, 17, ….\)
\(a = 3\) and \(d = a_2 − a_1 = 10 − 3 = 7\)
n^th term of this A.P. \(= 3 + (n − 1) 7\)
\(a_n = 3 + 7n − 7 \)
\(a_n = 7n − 4 \ \ \ \ ……(2)\)
It is given that, nth term of these A.P.s are equal to each other. Equating both these equations, we obtain
\(61 + 2n = 7n − 4\)
\(61 + 4 = 5n\)
\(5n = 65\)
\(n = 13\)

Therefore, 13^th terms of both these A.P.s are equal to each other.