Let two A.P's have the first terms a , a' and common difference d.
Given that difference between their 100th term is 100.
then a + 99d - a' - 99d = 100
⇒ a - a' = 100.
Now difference between 1000 th term is
⇒ a + 999d - a' - 999d
= a - a'
∴ difference between 1000 th term is 100.