Let <an> be an arithmetic progression satisfying a1 + a2 + a3 + ...... a100 = 100 and a101 + a102 + a103 + ...... a200 = 200. Then the common difference 'd' of arithmetic progression, is

Engineering

Mathematics

Arithmetic Progression and Its Properties

Question

Let <a_n> be an arithmetic progression satisfying

a₁ + a₂ + a₃ + ...... a₁₀₀ = 100 and a₁₀₁ + a₁₀₂ + a₁₀₃ + ...... a₂₀₀ = 200.

Then the common difference 'd' of arithmetic progression, is

$\frac{1}{1000}$

$\frac{1}{100}$

$\frac{1}{10}$

$\frac{1}{10000}$

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a₁ + a₂ + a₃ + ...... a₁₀₀ = 100

a + (a + d) + (a + 2d) +...... + (a + 99d) = 100

100a + d(1 + 2 + ....+ 99) = 100 .....(1)

a₁₀₁ + a₁₀₂ + a₁₀₃ + ...... a₂₀₀ = 200

(a + 100d) + (a + 101d) +....+ (a + 199d) = 200

100a + d(100 + 101 + ........ + 199)

= 200 .....(2)

(2) – (1)

⇒ d(100 + 100 + 100 + ....... + 100) = 100

= d (100)(100) = 100

$= d = \frac{1}{100} .$