Question

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

Answer 1

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.,
\(a_{100} = a_1 + (100 − 1)d\) \(= a_1 + 99d\)
\(a_{1000} = a_1 + (1000 − 1) d\) \(= a_1 + 999d\)
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.