Two APs have the same common difference. The difference between their 100th terms is 100, what is the difference between their 1000th terms?
The difference between their 1000th terms is 100.
Explanation
This question tests understanding that when two APs have the same common difference, the difference between any corresponding terms remains constant and equals the difference between their first terms. The context establishes that an AP is defined by first term 'a' and common difference 'd', and that consecutive terms differ by d. Using the nth term formula aₙ = a + (n-1)d, when we subtract corresponding terms of two APs with same d, the (n-1)d terms cancel out, leaving only the difference between first terms.
Solution Steps
Step 1: Let the first AP have first term 'a' and common difference 'd'. Let the second AP have first term 'a'' and the same common difference 'd'.
Step 2: The 100th term of first AP = a + 99d. The 100th term of second AP = a' + 99d.
Step 3: Difference between 100th terms = (a + 99d) - (a' + 99d) = a - a' = 100 (given).
Step 4: The 1000th term of first AP = a + 999d. The 1000th term of second AP = a' + 999d.
Step 5: Difference between 1000th terms = (a + 999d) - (a' + 999d) = a - a' = 100.