Two APs have the same common difference. The difference between their ...
The formula for nth term of an AP is aₙ = a + (n - 1) d
Here, aₙ is the nth term, a is the first term, d is the common difference and n is the number of terms.
For first A.P., a₁₀₀ = a₁ + (100 - 1) d
a₁₀₀ = a₁ + 99d ------ (1)
a₁₀₀₀ = a₁ + (1000 - 1)d
a₁₀₀₀ = a₁ + 999d ------ (2)
For second A.P.,
b₁₀₀ = b₁+ (100 - 1)d
b₁₀₀ = b₁ + 99d ------ (3)
b₁₀₀₀ = b₁ + (1000 - 1)d
b₁₀₀₀ = b₁ + 999d ------ (4)
Given that, difference between 100th term of these A.P.s = 100
Thus, from equations (1) and (3) we have
(a₁ + 99d ) - (b₁ + 99d ) = 100
a₁ - b₁ = 100 ...(5)
Difference between 1000th terms of these A.P.s
Thus, from equations (2) and (4) we have
(a₁ + 999d ) - (b₁ + 999d ) = a₁ - b₁
But a₁ - b₁ = 100 [From equation(5)]
Hence, the difference between the 1000th terms of these A.P. will be 100.