Question

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

Answer 1

For AP: $63, 65, 67, ….$
$a = 63$ and $d = a_2 − a_1 = 65 − 63 = 2$
nth 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 $d = a_2 − a_1 = 10 − 3 = 7$
nth 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, 13th terms of both these A.P.s are equal to each other.