Question

Two APs have the same common difference. The difference between their 100th terms is 100, what is the difference between their 1000th 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 100th term of these A.P.s = 100
Therefore,
$(a_1 + 99d) − (a_2 + 99d) = 100$
$a_1 − a_2 = 100 ……(1)$
Difference between 1000th 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 1000th terms of these A.P. will be 100.