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If a₁, a₂, ., a₄₀₀₁ are in arithmetic progression and 1 a₁ a₂ + 1 a₂ a₃ + . .+ 1 a₄₀₀₀ a₄₀₀₁ =10 and a₂+a₄₀₀₀=50 . Find the value of |a₁-a₄₀₀₁ | .

Correct answer

30

Step-by-step solution

aligned & 1 d ( a₂-a₁ a₂ a₁ + a₃-a₂ a₃ a₂ + . .+ a₄₀₀₁-a₄₀₀₀ a₄₀₀₁ a₄₀₀₀ )=10 &= 1 d ( 1 a₁ - 1 a₂ + 1 a₂ - 1 a₃ + .+ 1 a₄₀₀₀ - 1 a₄₀₀₁ )=10 &= 1 d ( 1 a₁ - 1 a₄₀₀₁ )=10 &= 1 d ( a₄₀₀₁-a₁ a₁ a₄₀₀₁ )=10 &= 4000 a₁ a₄₀₀₁ =10 ( as a₄₀₀₁=a₁+4000 d ) & a₁ a₄₀₀₁=400 ; a₂+a₄₀₀₀=50 (a₁+d )+ (a₁+3999 d )=50 & (a₁-a₄₀₀₁ )^2= (a₁+a₄₀₀₁ )^2-4 a₁ a₄₀₀₁ & |a₁-a₄₀₀₁ |=30 aligned

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