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

Question

Homi · Accepted Answer

The difference between any two corresponding terms of two APs with the same common difference is always constant.

Step 1 — Define the APs

Let's consider the first AP. We will call its first term $a_1$ . Its common difference is $d$ . The $n$ -th term of this AP is $A_n = a_1 + (n-1)d$ .

Now, let's consider the second AP. We will call its first term $b_1$ . Its common difference is also $d$ . The $n$ -th term of this AP is $B_n = b_1 + (n-1)d$ .

Diagram 1

Step 2 — Use the given information

We are told that the difference between their 100th terms is 100. Let's write down the 100th term for each AP.

$A_{100} = a_1 + (100-1)d$

$A_{100} = a_1 + 99d$

$B_{100} = b_1 + (100-1)d$

$B_{100} = b_1 + 99d$

Now, let's find the difference between these terms.

$A_{100} - B_{100} = (a_1 + 99d) - (b_1 + 99d)$

$= a_1 + 99d - b_1 - 99d$

$= a_1 - b_1$

We know this difference is 100.

$\boxed{a_1 - b_1 = 100}$

Step 3 — Find the difference for 1000th terms

We need to find the difference between their 1000th terms. Let's write down the 1000th term for each AP.

$A_{1000} = a_1 + (1000-1)d$

$A_{1000} = a_1 + 999d$

$B_{1000} = b_1 + (1000-1)d$

$B_{1000} = b_1 + 999d$

Now, let's find the difference between these terms.

$A_{1000} - B_{1000} = (a_1 + 999d) - (b_1 + 999d)$

$= a_1 + 999d - b_1 - 999d$

$= a_1 - b_1$

From Step 2, we found that $a_1 - b_1$ is 100. So, the difference between their 1000th terms is also 100.

$\boxed{A_{1000} - B_{1000} = 100}$

Answer

The difference between their 1000th terms is 100.

Question 12

Answer

More questions in Exercise 5.2