#### Question

Two arithmetic progressions have the same common difference. The difference between their 100th terms is 100, what is the difference between their l000th terms?

#### Solution

Here, we are given two A.P sequences which have the same common difference. Let us take the first term of one A.P. as *a* and of other A.P. as *a’*

Also, it is given that the difference between their 100^{th} terms is 100.

We need to find the difference between their 100^{th} terms

So, let us first find the 100^{th} terms for both of them.

Now, as we know,

`a_n = a + (n - 1)d`

So for 100 th term of first A.P (n = 100)

`a_100 = a + (100 - 1)d`

= a + 99d

Nw for 100 th term of second A.P (n = 100)

`a_100 = a + (100 - 1)d`

= a + 99d

Now, for 100 th term of second A.P (n = 100)

`a'_100 = a' + (100 - 1)d`

`= a' + 99d`

Now we are given

`a_100 - a'_100 = 100`

On substituting the values we get

a + 99d - a' - 99d = 100

a - a' = 100 .....(1)

Now we need the difference between the 1000 th term of both the A.P.

So for 1000 ^{th} term of second A.P. (n = 1000)

`a'_1000 = a' + (1000 - 1)d`

= a' + 999d

So,

`a_1000 - a'_1000 = (a + 999d) - (a' + 999d)`

=a + 999d - a' - 999d

= a- a'

= 100 (Using 1)

Therefore the difference ebetween the 1000 th terms of both the arithmetic progression will be 100