Let [ x ] denote greatest integer less than or equal to x. If for n∈N,
(1 - x + x³)ⁿ = Σⱼ₌₀³ⁿ aⱼxʲ,
then Σⱼ₌₀^([ 3n/2 ]) a₂ⱼ + 4 Σⱼ₌₀^([ (3n - 1)/2 ]) a₂ⱼ₊₁ is equal to :

Solution

Maths sir · 4 steps

Put x = 1 and x = -1: together they split the coefficients into even and odd sums.

1. Name the sums. Let E = a₀ + a₂ + a₄ + … be the sum of the even-index coefficients and O = a₁ + a₃ + … the sum of the odd-index ones. The expression asked for is E + 4O. The greatest-integer limits only make sure every coefficient up to a₃ₙ is included.
2. Put x = 1. (1 - 1 + 1)ⁿ = 1ⁿ = 1, so E + O = 1.
3. Put x = −1. Even powers of -1 give +1 and odd powers give -1. So (1 + 1 - 1)ⁿ = 1ⁿ = 1, which gives E - O = 1.

   E + O = 1, E - O = 1 ⇒ E = 1, O = 0

4. So. E + 4O = 1 + 4 × 0 = 1, which is option (d).

The idea to keep. Whenever a question separates even and odd coefficients, evaluate the polynomial at x = 1 and x = -1. Half their sum is the even part and half their difference is the odd part.

Answer from the NTA final answer key. Solution by PrepWiser.