Two APs have the same common difference. The difference between their 100th terms is 100, what is the difference between their 1000th terms?

Let the common difference of both the
A.P.’s = d
and First term of Ist A.P. = a
also first term of 2nd A .P. = b
Then, 200th term of 1st A .P. = a + 99d
and 100th term of IInd A .P. = b + 99d
Difference between their 100th term = 100
(given)
⇒ (a + 99d) – (b + 99d) = 100
⇒ a – b = 100 ...(i)
Now, a₁₀₀₀ of Ist A.P. = a + 999d
and a₁₀₀₀ of Ilnd A .P. = b + 999d
Hence, difference between their 1000th terms
= (a + 999d) (b + 999d) = a – b
= 100. [Using (i)]