Let ao, a1,...an, be real numbers. If as+ar+...+a," = (x+1)³ (x+2) ³... (x+672)³ determine a2 +a4+...+a2016
Answers
Given : a₀ , a₁ , a₂ ____aₙ real numbers.
a₀ + a₁x + a₂x² + a₃x³ + ______ + aₙxⁿ = (x+1)³ (x+2) ³____(x+672)³
To Find : a₂ + a₄ + + ______ + a₂₀₁₆
Solution:
a₀ + a₁x + a₂x² + a₃x³ + ______ + aₙxⁿ = (x+1)³ (x+2) ³____(x+672)³
x = 0
=> a₀ = 1³ * 2³ * 3³ _______ * 672³ = (672 !)³
x = 1
=> a₀ + a₁ + a₂ + a₃ + ______ + a₂₀₁₆ = (2)³ (3) ³____( 673)³ = (673!)³
x = - 1
=> a₀ - a₁ + a₂ - a₃ + ______ + a₂₀₁₆ = (0)³ (1) ³____( 671)³ = 0
Adding both
=> 2 ( a₀ + a₂ + a₄ + + ______ + a₂₀₁₆ ) = (673!)³
a₀ + a₂ + a₄ + + ______ + a₂₀₁₆ = (673!)³ / 2
a₀ = (672 !)³
=> a₂ + a₄ + + ______ + a₂₀₁₆ = (673!)³ /2 - (672 !)³
=> a₂ + a₄ + + ______ + a₂₀₁₆ = (672 !)³ ( 673³ /2 - 1)
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