GIVEN :-

• Two APs have the same common difference.
• The difference between their 100th terms is 100.

To Find :-

• what is the difference between their 1000TH terms ?

Formula used :-

• nth Term of AP = a + ( n - 1)d

Solution :-

→ Lets First Term of First AP = a

→ Let Common Difference of First AP = d

So,

→ T₁₀₀ = a + (100 - 1)d

→ T₁₀₀ = a + 99d ------------ Equation (1)

→ Lets First Term of Second AP = b

→ Let Common Difference of Second AP = d

So,

→ T₁₀₀ = b + (100 - 1)d

→ T₁₀₀ = b + 99d ------------ Equation (2)

Given That, Equation (1) - Equation (2) = 100 .

So,

→ ( a + 99d) - (b + 99d) = 100

→ a - b + 99d - 99d = 100

→ (a - b) = 100 ------------------ Equation (3). .

_________________

Now, we have to Find difference between their 1000th terms .

So,

→ T₁₀₀₀ = a + (1000 - 1)d ( First Series) .

→ T₁₀₀₀ = a + 999d ---------- Equation (4)

And,

→ T₁₀₀₀ = b + (1000 - 1)d (Second Series)

→ T₁₀₀₀ = b + 999d ----------------- Equation (5) .

So, Equation (4) - Equation (5) :-

→ (a + 999d) - (b + 999d)

→ (a - b) + (999d - 999d)

→ (a - b) + 0

→ (a - b)

PUTTING Value of Equation (3) Here , we get,

→ 100 (ANS.)