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

Solution:

Let, the first term of two APs be a₁ and A₁ respectively.

The common difference of these two APs is d.

For the first AP, we know,

a_n = a + (n − 1) d

Therefore,

a₁₀₀ = a₁ + (100 − 1) d

= a₁ + 99d

a₁₀₀₀ = a₁ + (1000 − 1) d

a₁₀₀₀ = a₁ + 999 d

For second A.P., we know,

A_n = A + (N − 1)d

Therefore,

A₁₀₀ = A₂ + (100 − 1)d

= A₂ + 99d

A₁₀₀₀ = A₂ + (1000 − 1)d

= A₂ + 999d

Given that, difference between 100^th term of the two APs = 100

Therefore, ( a₁ + 99d) − (A₁ + 99 d) = 100

a₁ − A₁ = 100……………………………………………………………….. (i)

Difference between the 1000^th terms of the two APs

(a₁ + 999 d ) − (A₁ + 999d) = a₁ − A₁

From equation (i),

This difference, a₁ − A₁ = 100

Hence, the difference between the 1000^th terms of the two APs is 100.