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'

= 100

∴ difference between 1000 th term is 100.