Question

Question 15
For what value of n, are the $n^{th}$ terms of two APs 63, 65, 67, and 3, 10, 17, … equal?

Answer 1

$63,65,67,\dots \dots a=63d=a_2-a_1=65-63=2n^{th}termofthisA.P,a_n=a+(n-1)d{a}_n=63+(n-1)2=63+2n-2a_n=61+2n\dots \dots \left(i\right)3,10,17\dots \dots a=3d=a_2-a_1=10-3=7n^{th}termofthisA.P=3+(n-1)7a_n=3+7n-7a_n=7n-4\dots \dots \left(ii\right)$
It is given that, $n^{th}$ term of these A.Ps 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.Ps are equal to each other.