The function given is:

$f(n) = \left( \frac{1}{2} + \frac{4}{5}i \right)^n$

This expression involves a complex number raised to the power of $n$, where the complex number is $z = \frac{1}{2} + \frac{4}{5}i$. The task is to observe how the graph changes as $n$ increases, starting from $n = 1$.

Step-by-step analysis:

1. Complex Number Representation:

   • $z = \frac{1}{2} + \frac{4}{5}i$ can be expressed in polar form. First, find the magnitude and argument of $z$:
     • Magnitude: $|z| = \sqrt{\left(\frac{1}{2}\right)^2 + \left(\frac{4}{5}\right)^2} = \sqrt{\frac{1}{4} + \frac{16}{25}} = \sqrt{\frac{41}{100}} = \frac{\sqrt{41}}{10}$.
     • Argument: $\theta = \tan^{-1}\left(\frac{4/5}{1/2}\right) = \tan^{-1}(1.6)$.

2. Raising to Powers:

   • When a complex number is raised to a power, its magnitude gets raised to that power, and the argument (angle) is multiplied by that power.
   • Therefore, as $n$ increases, the magnitude will grow or shrink depending on $|z|^n$, and the points rotate around the origin in the complex plane according to the argument $n \times \theta$.

3. Graph Behavior:

   • The graphs shown below the equation represent how the points rotate and spiral as $n$ increases. The blue dots represent the points plotted after each iteration, and the red arrows indicate the direction of movement.
   • In each case, the function creates a rotational pattern as the value of $n$ increases, creating spirals or polygons depending on the magnitude and angle.

4. Effect of $n$:

   • For small values of $n$, the graph shows a shape that seems to resemble a polygonal path. As $n$ increases, the points become more tightly packed, and the path becomes increasingly complex.

Would you like me to go further into the mathematical details of how the graph evolves or explain the transformation from rectangular to polar coordinates more deeply?

Related Questions:

1. How can you find the polar form of any complex number?
2. Why does raising a complex number to a power result in spirals or rotational shapes?
3. How does the magnitude of the complex number $|z|$ affect the shape of the graph as $n$ increases?
4. What is the significance of the argument $\theta$ in the context of the graph?
5. How do you calculate the number of "vertices" or points visible in the polygon for small $n$?

Tip:

When working with complex numbers in exponential or polar form, keep track of both magnitude and phase (angle) to understand the behavior of the resulting curves or spirals.