Multiple choice

State whether the following statement is true or false. If $\beta$ is an imaginary root of the equation $z^n \, - \, 1 \, = \, 0$, then $1 \, + \, \beta \, + \, \beta^2 \, + \, ..... \, + \, \beta^{n \, - \, 1} \, = \, 0$

  1. True

  2. False
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A Correct answer
Explanation

The expression 1 + beta + beta^2 + ... + beta^(n-1) is a geometric series sum equal to (beta^n - 1) / (beta - 1). Since beta is a root of z^n - 1 = 0, beta^n = 1. Thus, the sum is 0 / (beta - 1) = 0.

AI explanation

If beta is an imaginary root of the equation z^n - 1 = 0, it must be true that beta^n = 1. The geometric series sum 1 + beta + beta^2 + ... + beta^(n-1) can be simplified using the formula (beta^n - 1) / (beta - 1). Substituting beta^n = 1 into the numerator of this fraction results in (1 - 1) / (beta - 1) = 0. Because beta is imaginary, it is not equal to 1, confirming the sum is exactly zero.

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