Question

find the graph of this function as the value of n increases, starting from n = 1.

f(n) = (\frac{3}{5} + \frac{4}{5}i)^n

remember: |a + bi| = \sqrt{a^2 + b^2}

Explanation:

Step1: Calculate the modulus of the complex number

For the complex number \(z=\frac{3}{5}+\frac{4}{5}i\), using the formula \(|a + bi|=\sqrt{a^{2}+b^{2}}\), we have \(|z|=\sqrt{(\frac{3}{5})^{2}+(\frac{4}{5})^{2}}=\sqrt{\frac{9 + 16}{25}} = 1\).

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

Since \(|z| = 1\), by De - Moivre's theorem \(z=\cos\theta+i\sin\theta\) (where \(\cos\theta=\frac{3}{5}\), \(\sin\theta=\frac{4}{5}\)), and \(z^{n}=\cos(n\theta)+i\sin(n\theta)\). The modulus of \(z^{n}\) is \(|z^{n}|=|z|^{n}=1^{n} = 1\). As \(n\) increases, the points \(z^{n}\) lie on the unit circle centered at the origin.

Answer:

The second graph (the one where all the points lie on a unit - circle centered at the origin) is the correct graph.