Question

For what value of n, are the n^th terms of two APs 63, 65, 67, and 3, 10, 17, … equal

Answer 1

63, 65, 67, …

a = 63

d = a₂ − a₁ = 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)

3, 10, 17, …

a = 3

d = a₂ − a₁ = 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, n^th 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.