Question

Let ao, a1,...an, be real numbers. If as+ar+...+a," = (x+1)³ (x+2) ³... (x+672)³ determine a2 +a4+...+a2016​

Answer 1

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)