For what value of n, are the nth terms of two AP’s: 63, 65, 67… and 3, 10, 17… equal?

Lets first consider AP 63, 65, 67…

First term = a = 63,
Common difference = d
= 65 – 63 = 2

Using formula , to find \(n^{th}\) term of arithmetic progression,

\(a_n\) = 63 + (n - 1) (2)… (1)

Now, consider second AP 3, 10, 17…
First term = a = 3,
Common difference = d
= 10 – 3 = 7
Using formula , to find \(n^{th}\) term of arithmetic progression,

\(a_n\) = 3 + (n - 1) (7)… (2)

According to the given condition:

\(\Rightarrow\) (1) = (2)
\(\Rightarrow \) 63 + (n - 1) (2) = 3 + (n - 1) (7)
\(\Rightarrow \) 63 + 2n – 2 = 3 + 7n - 7
\(\Rightarrow \) 65 = 5n
\(\Rightarrow \) n = 13

Therefore, \(13^{th}\) terms of both the AP’s are equal.