Two AP’s have the same common difference. The difference between their \(100^{th}\) terms is 100, what is the difference between their \(1000^{th}\) terms.

Let first term of \(1^{st}\) AP = a
Let first term of \(2^{nd}\) AP = a'

It is given that their common difference is same.

Let their common difference be d.

It is given that difference between their \(100^{th}\) terms is 100.

Using formula \(a_n = a + (n - 1)d\), to find nth term of arithmetic progression,

\(\Rightarrow \) a + (100 - 1) d – [a' + (100 - 1) d]= 100
\(\Rightarrow \) a + 99d - a' - 99d = 100
\(\Rightarrow \) a - a' = 100… (1)

We want to find difference between their \(1000^{th}\) terms which means we want to calculate:

a + (1000 - 1) d – [a' + (1000 - 1) d]
= a + 999d - a' - 999d = a – a'

Putting equation (1) in the above equation,

a + (1000 - 1) d – [a' + (1000 - 1) d]
= a + 999d - a' + 999d = a - a' = 100

Therefore, difference between their \(1000^{th}\) terms would be equal to 100.