Show ω⁷=1 and deduce 1+ω+⋯+ω⁶=0.

01 / Standard#Your turn

\omega=\cos\frac{2\pi}7+i\sin\frac{2\pi}7

Official paper · jm02-2022 · 4(b)(i) · PDF 6

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Apply De Moivre.

Hint 2

Use the finite geometric sum and verify ω≠1.

Worked solution

Raise the angle to seven times its value.
$\omega^7=\cos2\pi+i\sin2\pi=1$
Since 2π/7 is not a multiple of 2π, ω−1≠0.
$1+\omega+\cdots+\omega^6=\frac{\omega^7-1}{\omega-1}=0$

ω⁷=1; the seven-term sum is zero.

Checks and common pitfalls: Division by ω−1 needs the explicit ω≠1 check.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Raise the angle to seven times its value.
$\omega^7=\cos2\pi+i\sin2\pi=1$
Since 2π/7 is not a multiple of 2π, ω−1≠0.
$1+\omega+\cdots+\omega^6=\frac{\omega^7-1}{\omega-1}=0$

Think first. Reveal a hint when the class is ready.

2022 JM02