If z² + z + 1 = 0, z ∈ C, then
| Σₙ₌₁¹⁵ ( zⁿ + ( - 1)ⁿ1/zⁿ )² | is equal to _________.

Solution

Maths sir · 4 steps

Here z is ω, so expand the square and let the powers repeat every 3 steps.

1. Identify z. The roots of z² + z + 1 = 0 are ω and ω², the non-real cube roots of unity. So z³ = 1 and 1 + z + z² = 0.

   (zⁿ + (-1)ⁿ z⁻ⁿ)² = z²ⁿ + z⁻²ⁿ + 2(-1)ⁿ

2. Power part. Since z³ⁿ = 1, we get z⁻²ⁿ = zⁿ, so z²ⁿ + z⁻²ⁿ = z²ⁿ + zⁿ. This equals 2 when 3 divides n and -1 otherwise. From 1 to 15 there are 5 multiples of 3, so the total is 5 × 2 + 10 × (-1) = 0.
3. Sign part. From n = 1 to 15 there are 8 odd values and 7 even values, so Σ (-1)ⁿ = -8 + 7 = -1. The middle terms add up to 2 × (-1) = -2.
4. So. The whole sum is 0 + (-2) = -2, and its modulus is 2.

The common slip: 30. Taking the cross term as +2 for every n and ignoring the (-1)ⁿ gives 30, not 2.

The idea to keep. Whenever z² + z + 1 = 0, replace z by ω. Then ωⁿ + ω²ⁿ is 2 if 3 divides n and -1 otherwise, so long sums collapse in groups of three.

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