Question

find the graph of this function as the value of n increases, starting from n = 1.
f(n) = (\frac{12}{17}+\frac{13}{17}i)^n
remember: |a + bi| = \sqrt{a^2 + b^2}

Explanation:

Step1: Calculate the modulus of the complex - number

First, find the modulus of the complex number $z=\frac{12}{17}+\frac{13}{17}i$. Using the formula $|a + bi|=\sqrt{a^{2}+b^{2}}$, we have $|z|=\sqrt{(\frac{12}{17})^{2}+(\frac{13}{17})^{2}}=\sqrt{\frac{144 + 169}{289}}=\sqrt{\frac{313}{289}}\approx1.04$.

Step2: Analyze the behavior of $f(n)$ as $n$ increases

When $n = 1$, $f(1)=\frac{12}{17}+\frac{13}{17}i$. As $n$ increases, since $|z|\approx1.04>1$, the magnitude of $f(n)=z^{n}$ will increase. The points representing $f(n)$ in the complex - plane will move further away from the origin as $n$ increases. Also, since $z$ is a non - real complex number, the points will be arranged in a circular - like pattern (because of the argument of the complex number).

Answer:

The points representing $f(n)$ in the complex - plane will move further away from the origin as $n$ increases and will be arranged in a circular - like pattern.