Two APs have the same common difference. The difference between their 100th term is 100, what is the difference between their 1000thterms?
Let the first term of these A.P.s be a1 and a2 respectively and the common difference of these A.P.s be d.
For first A.P.,
a100 = a1 + (100 − 1) d
= a1 + 99d
a1000 = a1 + (1000 − 1) d
a1000 = a1 + 999d
For second A.P.,
a100 = a2 + (100 − 1) d
= a2 + 99d
a1000 = a2 + (1000 − 1) d
= a2 + 999d
Given that, difference between
100th term of these A.P.s = 100
Therefore, (a1 + 99d) − (a2 + 99d) = 100
a1 − a2 = 100 (1)
Difference between 1000th terms of these A.P.s
(a1 + 999d) − (a2 + 999d) = a1 − a2
From equation (1),
This difference, a1 − a2 = 100
Hence, the difference between 1000th terms of these A.P. will be 100.