The given function is:

$f(n) = \left( \frac{7}{13} + \frac{6}{7}i \right)^n$

This expression involves a complex number raised to the power of $n$, where $i$ is the imaginary unit. Let's break this down:

1. Complex Number Representation: The complex number here is $z = \frac{7}{13} + \frac{6}{7}i$. This number can be written in the form $z = a + bi$, where $a = \frac{7}{13}$ and $b = \frac{6}{7}$.

2. Modulus and Argument: To analyze the behavior of $f(n)$ as $n$ increases, we can compute the modulus and argument (polar form) of $z$:

   • The modulus $|z|$ is given by $|z| = \sqrt{a^2 + b^2}$.
   • The argument $\theta$ is the angle of the complex number in the complex plane, given by $\theta = \tan^{-1}\left(\frac{b}{a}\right)$.

3. Repeated Powers of Complex Numbers: When raising a complex number to a power $n$, its modulus gets raised to the power $n$ and the argument gets multiplied by $n$. This means that the magnitude of the resulting number depends on $|z|^n$, and the angle rotates by $n \theta$ as $n$ increases.

Visual Interpretation:

• The images at the bottom show a pattern that resembles spirals or rotations in the complex plane. As $n$ increases, the points follow a path that increasingly circles around the origin.
• These spirals result from the repeated multiplication of the complex number by itself, which creates a rotation (due to the argument $\theta$) and a possible shrinking or expanding (depending on the modulus $|z|$).

Behavior as $n$ Increases:

• The modulus $|z|$ will determine whether the points move towards the origin or away from it. If $|z| < 1$, the points spiral inward, and if $|z| > 1$, they spiral outward.
• The rotation is governed by the argument $\theta$, creating circular patterns in the complex plane as seen in the provided graphs.

Would you like more details on any part of this analysis or have any specific questions about the function?

Here are five related questions:

1. How is the polar form of a complex number used in analyzing powers of complex numbers?
2. What is the significance of the modulus $|z|$ when raising a complex number to a power?
3. How does the argument $\theta$ affect the rotation of points in the complex plane?
4. What happens to the shape of the graph if $n$ increases indefinitely?
5. How do the real and imaginary parts of $z$ influence the structure of the graph?

Tip: When analyzing powers of complex numbers, the modulus tells you about the scaling (shrinking or expanding) effect, while the argument controls the rotation.