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

#### Solution

Let the first term of these A.P.s be a_{1} and a_{2} respectively and the common difference of these A.P.s be d.

For first A.P.,

a_{100} = a_{1} + (100 − 1) d

= a_{1} + 99d

a_{1000} = a_{1} + (1000 − 1) d

a_{1000} = a_{1} + 999d

For second A.P.,

a_{100} = a_{2} + (100 − 1) d

= a_{2} + 99d

a_{1000} = a_{2} + (1000 − 1) d

= a_{2} + 999d

Given that, difference between

100^{th} term of these A.P.s = 100

Therefore, (a_{1} + 99d) − (a_{2} + 99d) = 100

a_{1} − a_{2} = 100 (1)

Difference between 1000^{th} terms of these A.P.s

(a_{1} + 999d) − (a_{2} + 999d) = a_{1} − a_{2}

From equation (1),

This difference, a_{1} − a_{2 }= 100

Hence, the difference between 1000^{th} terms of these A.P. will be 100.