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

Let first term of first A.P. be a₁

and

first term of second A.P. be b₁.

Let common difference be d.

For 1st A.P. :

100th term = a₁ + (100 - 1)d

= a₁ + 99d.

For 2nd A.P. :

100th term = b₁ + (100 - 1)d

= b₁ + 99d.

Given,

Difference between 100th terms is 100 :

∴ b₁ + 99d - (a₁ + 99d) = 100

⇒ b₁ - a₁ + 99d - 99d = 100

⇒ b₁ - a₁ = 100 ……….(1)

For 1st A.P. :

1000th term = a₁ + (1000 - 1)d

= a₁ + 999d.

For 2nd A.P. :

1000th term = b₁ + (1000 - 1)d

= b₁ + 999d.

Difference between 1000th terms :

⇒ b₁ + 999d - (a₁ + 999d)

⇒ b₁ - a₁ + 999d - 999d

⇒ b₁ - a₁

⇒ 100 [From (1)]

Hence, difference between 1000 terms of the A.P. will be 100.