Question

find the graph of this function as the value of n increases, starting from n = 1.
$f(n) = (\frac{8}{17}+\frac{15}{17}i)^n$
remember: $|a + bi| = \sqrt{a^{2}+b^{2}}$

Explanation:

Step1: Calculate the modulus of the complex number

The modulus of the complex number \(z=\frac{8}{17}+\frac{15}{17}i\) is \(|z|=\sqrt{(\frac{8}{17})^2 + (\frac{15}{17})^2}\).

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Step2: Analyze the modulus of \(f(n)\)

Since \(f(n)=z^n\) and \(|z| = 1\), then \(|f(n)|=|z^n|=|z|^n\). Substituting \(|z| = 1\) into \(|z|^n\), we get \(|f(n)|=1^n=1\) for all \(n\in N\).

Answer:

The third graph (the one where the points lie on a unit - circle centered at the origin) is the correct graph. As \(n\) increases, the points \(f(n)\) lie on the unit circle \(x^{2}+y^{2}=1\) because \(|f(n)| = 1\) for all \(n\).