Question

find the graph of this function as the value of n increases, starting from n = 1. f(n) = (5/13 + 12/13 i)^n remember |a + bi| = √(a² + b²)

Explanation:

Step1: Find 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 power of the complex number

When we raise a complex number \( z \) with \( |z| = 1 \) to the power \( n \), i.e., \( z^{n}=(\frac{5}{13}+\frac{12}{13}i)^{n} \), the modulus of \( z^{n} \) is \( |z|^{n}=1^{n} = 1 \) for all positive integers \( n \). This means that all the points \( f(n)=z^{n} \) lie on the unit circle (a circle with radius 1 centered at the origin) in the complex plane.

Now, let's check the graphs:

• The first graph: The points seem to be on a circle but the connections and distribution look different from a regular unit - circle - based power of a complex number with \( |z| = 1 \).
• The second graph: The points are on a circle (the unit circle) and as \( n \) increases, since \( z=\frac{5}{13}+\frac{12}{13}i=\cos\theta+i\sin\theta \) where \( \cos\theta=\frac{5}{13} \) and \( \sin\theta=\frac{12}{13} \) (because \( \cos^{2}\theta+\sin^{2}\theta = 1 \) and \( \theta=\arctan(\frac{12/13}{5/13})=\arctan(\frac{12}{5}) \), a fixed angle), \( z^{n}=\cos(n\theta)+i\sin(n\theta) \), so the points will be evenly (in terms of angular spacing) distributed on the unit circle as \( n \) increases. This graph shows points on the unit circle.
• The third graph: The points are not on a circle (they seem to be clustered near the origin), so this cannot be the graph of \( f(n) \) since \( |f(n)| = 1 \) for all \( n \).

Answer:

The Middle Graph