Question

find the graph of this function as the value of n increases, starting from n = 1. f(n) = (1/2 + 4/5 i)^n remember |a + bi| = √(a² + b²)

Explanation:

Step1: Calculate the modulus of the complex number

The complex number is \( z = \frac{1}{2}+\frac{4}{5}i \). Using the formula \( |a + bi|=\sqrt{a^{2}+b^{2}} \), we have \( a=\frac{1}{2} \) and \( b = \frac{4}{5} \). Then \( |z|=\sqrt{(\frac{1}{2})^{2}+(\frac{4}{5})^{2}}=\sqrt{\frac{1}{4}+\frac{16}{25}}=\sqrt{\frac{25 + 64}{100}}=\sqrt{\frac{89}{100}}\approx0.943 \), which is less than 1.

Step2: Analyze the behavior as \( n \) increases

For a complex number \( z \) with \( |z|=r \), when we take \( z^{n} \), the modulus of \( z^{n} \) is \( |z|^{n}=r^{n} \). Since \( r\approx0.943< 1 \), as \( n \) increases, \( r^{n} \) will decrease (because a number between 0 and 1 raised to a larger power gets smaller). So the points representing \( f(n)=(\frac{1}{2}+\frac{4}{5}i)^{n} \) should be getting closer to the origin as \( n \) increases.

Looking at the three graphs:

• The first graph: The points seem to be on a circle with radius around 1 (or not decreasing in distance from origin).
• The second graph: The points are getting closer to the origin as \( n \) increases (since the radius of the "circle" of points is decreasing).
• The third graph: The points seem to be on a circle or moving away, not towards the origin.

So the middle graph (the second one) is the correct one as it shows the points getting closer to the origin (since the modulus is less than 1, the magnitude of \( z^{n} \) decreases with \( n \)).

Answer:

The Middle Graph