Question

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

Answer 1

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

Now, 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.s

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 between the 1000^th terms of both the arithmetic progressions will be 100.