If α is a root of the equation x²+x+1=0 and Σk=1ⁿ(αᵏ+1/αᵏ)²=20, then n is equal to _________.

Solution

Maths sir · 4 steps

Since 1/ω = ω², each bracket is ωᵏ + ω²ᵏ, and that repeats every 3 terms.

1. Identify alpha. The roots of x² + x + 1 = 0 are ω and ω², the non-real cube roots of unity. Take α = ω. The other root gives the same terms. Since ω³ = 1, we get 1/ωᵏ = ω²ᵏ.

   (ωᵏ + 1/ωᵏ)² = (ωᵏ + ω²ᵏ)²

2. Value of each term. If k is not a multiple of 3, then ωᵏ + ω²ᵏ = ω + ω² = -1, so the term is 1. If k is a multiple of 3, then ωᵏ + ω²ᵏ = 1 + 1 = 2, so the term is 4.
3. Add in blocks. Every block of three terms (k = 1, 2, 3) adds up to 1 + 1 + 4 = 6. So the first 9 terms give 3 × 6 = 18.
4. So. Terms k = 10 and k = 11 are not multiples of 3, so each adds 1. That gives 18 + 1 + 1 = 20, so n = 11.

The common slip: 20. Treating every term as (-1)² = 1 and forgetting that multiples of 3 give 2² = 4 gives 20, not 11.

The idea to keep. With powers of ω, always split by k mod 3. The terms repeat in a cycle of three, so sum one full cycle first and then add the leftover terms one by one.

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