Question

find the graph of this function as the value of n increases, starting from n = 1. f(n)=\left(\frac{5}{13}+\frac{12}{13}i\
ight)^n remember: |a + bi|=\sqrt{a^2 + b^2}

Explanation:

Step1: Calculate the modulus of the complex number

For the complex number \(z=\frac{5}{13}+\frac{12}{13}i\), using the formula \(|a + bi|=\sqrt{a^{2}+b^{2}}\), we have \(|z|=\sqrt{(\frac{5}{13})^{2}+(\frac{12}{13})^{2}}=\sqrt{\frac{25 + 144}{169}}=\sqrt{\frac{169}{169}} = 1\).

Step2: Analyze the property of \(f(n)=z^{n}\)

Since \(|z| = 1\), for \(f(n)=z^{n}\), we know that \(|f(n)|=|z^{n}|=|z|^{n}=1^{n}=1\). As \(n\) increases, the points representing \(f(n)\) on the complex - plane lie on the unit circle \(x^{2}+y^{2}=1\).

Answer:

The third graph (the one with points lying on a circle centered at the origin with radius \(1\)) is the correct graph.