Let the first A.P be:

a, a + d, a + 2d, ……..

Second A.P be:

b, b + d, b + 2d, b + 3d, ………

Also, general term, a_n = a + (n – 1)d

Given that, a₁₀₀ – b₁₀₀ = 100

⇒ a + d – (b + d) =

⇒ a – b =

Now the difference between their 1000th terms,

a₁₀₀₀ = a + d

b₁₀₀₀ = b + d

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(-)     a - b

∴ The difference between their 1000^th terms is (a – b) = .

Note: If the common difference for any two A.Ps are equal then difference between n^th terms of two A.Ps is same for all natural values of n.