Question

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

Explanation:

Step1: Find the modulus of the complex - number

First, for the complex number $z=\frac{5}{13}+\frac{12}{13}i$, use the formula $|a + bi|=\sqrt{a^{2}+b^{2}}$. Here, $a = \frac{5}{13}$ and $b=\frac{12}{13}$, so $|z|=\sqrt{(\frac{5}{13})^{2}+(\frac{12}{13})^{2}}=\sqrt{\frac{25 + 144}{169}}=\sqrt{\frac{169}{169}} = 1$.

Step2: Recall De - Moivre's theorem

De - Moivre's theorem states that for a complex number $z = r(\cos\theta+i\sin\theta)$ and positive integer $n$, $z^{n}=r^{n}(\cos(n\theta)+i\sin(n\theta))$. Since $r = |z| = 1$ for $z=\frac{5}{13}+\frac{12}{13}i$, and $\cos\theta=\frac{5}{13}$, $\sin\theta=\frac{12}{13}$, then $f(n)=(\frac{5}{13}+\frac{12}{13}i)^{n}=\cos(n\theta)+i\sin(n\theta)$ where $\theta=\arctan(\frac{12}{5})$.
As $n$ increases from $n = 1$, the points $f(n)$ on the complex - plane lie on the unit circle $|z| = 1$. The values of $f(n)$ are the $n$ - th power of the complex number $\frac{5}{13}+\frac{12}{13}i$. Each successive value of $n$ corresponds to a rotation of the complex number around the unit circle.
The points $f(n)$ will form a set of points evenly spaced around the unit circle as $n$ increases.

Answer:

The points $f(n)$ lie on the unit circle $|z| = 1$ and are evenly spaced around it as $n$ increases.