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

Solution:

The formula for n^th term of an AP is aₙ = a + (n - 1) d

Here, aₙ is the n^th 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 100^th 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 1000^th 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 1000^th terms of these A.P. will be 100.

☛ Check: NCERT Solutions for Class 10 Maths Chapter 5

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