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