Complex Numbers
Cube Roots of Unity and Geometric Series
Grade 11
Question:
<p>Let <i>w</i> be a non real cube root of unity then the number of distinct elements in the set <i>{(1 + w + w<sup>2</sup> + ... + w<sup>n</sup>)<sup>m</sup>; n, m ∈ ℕ}</i> is:</p>
Step-by-Step Solution
Key Concept: Use the property 1 + w + w² = 0 for non-real cube roots of unity to determine all possible values of the geometric series sum and their powers.
<p><strong>Analysis:</strong> Since <i>w</i> is a non-real cube root of unity, we have <i>w³ = 1</i> and <i>1 + w + w² = 0</i>.</p><p>The sum <i>1 + w + w² + ... + w<sup>n</sup></i> depends on <i>n mod 3</i>:</p><ul><li>If <i>n ≡ 0 (mod 3)</i>: sum = <i>1 + (1 + w + w²) + ... = 1</i> (when <i>n = 0</i>) or cycles through values</li><li>If <i>n ≡ 1 (mod 3)</i>: sum = <i>1 + w</i></li><li>If <i>n ≡ 2 (mod 3)</i>: sum = <i>1 + w + w²</i> = 0</li></ul><p>When the sum equals 0, the <i>m</i>-th power is 0. Otherwise, raising non-zero values to powers of <i>m</i> yields distinct cube roots.</p><p>∴ The number of distinct elements is <b>7</b>.</p>
Correct Answer: S