Two APs have the same common difference. The difference between their 100th terms is 100. What is the difference between their 1000th terms?
100
Step 1: Write general terms of the two APs.
Let the first AP have first term \(a_1\) and common difference \(d\).
Let the second AP have first term \(b_1\) and the same common difference \(d\).
Then the nth terms are:
First AP: \(T_n^{(1)} = a_1 + (n - 1)d\)
Second AP: \(T_n^{(2)} = b_1 + (n - 1)d\)
Step 2: Use the condition about the 100th terms.
Difference between their 100th terms is 100:
\[T_{100}^{(1)} - T_{100}^{(2)} = 100\]
Substitute the expressions:
\[(a_1 + 99d) - (b_1 + 99d) = 100\]
Simplify:
\[a_1 + 99d - b_1 - 99d = 100\]
\[a_1 - b_1 = 100\]
Step 3: Use this for the 1000th terms.
Now consider the difference between their 1000th terms:
\[T_{1000}^{(1)} - T_{1000}^{(2)}\]
Using the formula:
First AP: \(T_{1000}^{(1)} = a_1 + 999d\)
Second AP: \(T_{1000}^{(2)} = b_1 + 999d\)
So,
\[T_{1000}^{(1)} - T_{1000}^{(2)} = (a_1 + 999d) - (b_1 + 999d)\]
\[= a_1 + 999d - b_1 - 999d\]
\[= a_1 - b_1\]
From Step 2, we already know that:
\[a_1 - b_1 = 100\]
Conclusion: The difference between their 1000th terms is also 100.