Question

find the graph of this function as the value of n increases, starting from n = 1. f(n) = (14/15 + 4/15 i)^n remember: |a + bi| = √(a² + b²)

Explanation:

Step1: Find the modulus of the complex number

The complex number is \( z=\frac{14}{15}+\frac{4}{15}i \). Using the formula \( |a + bi|=\sqrt{a^{2}+b^{2}} \), we have \( a=\frac{14}{15} \) and \( b = \frac{4}{15} \). So \( |z|=\sqrt{(\frac{14}{15})^{2}+(\frac{4}{15})^{2}}=\sqrt{\frac{196 + 16}{225}}=\sqrt{\frac{212}{225}}=\frac{\sqrt{212}}{15}\approx\frac{14.56}{15}\approx0.97 \), which is less than 1? Wait, no, wait: \( 14^2=196 \), \( 4^2 = 16 \), sum is 212? Wait, no, wait, the complex number is \( \frac{14}{15}+\frac{4}{15}i \), so \( a=\frac{14}{15} \), \( b=\frac{4}{15} \). Then \( |z|=\sqrt{(\frac{14}{15})^2+(\frac{4}{15})^2}=\sqrt{\frac{196 + 16}{225}}=\sqrt{\frac{212}{225}}=\frac{\sqrt{212}}{15}\approx\frac{14.56}{15}\approx0.97 \)? Wait, no, wait, 14 squared is 196, 4 squared is 16, sum is 212? Wait, no, 14^2 is 196, 4^2 is 16, 196+16=212? Wait, but 15^2 is 225. Wait, but maybe I made a mistake. Wait, the modulus of a complex number \( z = r(\cos\theta + i\sin\theta) \) is \( r \), and when we raise it to the nth power, we get \( z^n = r^n(\cos(n\theta) + i\sin(n\theta)) \). So first, let's find the modulus \( r \) of \( \frac{14}{15}+\frac{4}{15}i \). \( r=\sqrt{(\frac{14}{15})^2+(\frac{4}{15})^2}=\sqrt{\frac{196 + 16}{225}}=\sqrt{\frac{212}{225}}=\frac{\sqrt{212}}{15}\approx\frac{14.56}{15}\approx0.97 \)? Wait, no, wait, 14^2 is 196, 4^2 is 16, 196+16=212? Wait, but 15^2 is 225. Wait, but maybe I miscalculated. Wait, 14^2 is 196, 4^2 is 16, sum is 212. Then sqrt(212) is approx 14.56. So 14.56/15 ≈ 0.97, which is less than 1? Wait, no, wait, 14^2 +4^2=196+16=212, yes. But wait, maybe the problem is that the modulus is less than 1? Wait, no, wait, 14/15 is about 0.933, 4/15 is about 0.266. Then the modulus is sqrt(0.933^2 + 0.266^2) ≈ sqrt(0.87 + 0.07) = sqrt(0.94) ≈ 0.97, which is less than 1. Wait, but if the modulus is less than 1, then as n increases, \( r^n \) approaches 0. But the graphs: the first graph is a spiral towards the origin, the second is a circle? Wait, no, wait, maybe I made a mistake. Wait, no, the complex number is \( \frac{14}{15}+\frac{4}{15}i \), let's write it in polar form. Let \( r = \sqrt{(\frac{14}{15})^2 + (\frac{4}{15})^2} = \frac{\sqrt{14^2 + 4^2}}{15} = \frac{\sqrt{196 + 16}}{15} = \frac{\sqrt{212}}{15} \). Wait, but 14^2 +4^2=212, but 15^2=225, so 212 < 225, so r < 1. Wait, but then when we raise to the nth power, the modulus is \( r^n \), which decreases as n increases (since r < 1). The argument \( \theta \) is \( \arctan(\frac{4/15}{14/15}) = \arctan(\frac{4}{14}) = \arctan(\frac{2}{7}) \), which is a constant. So the points \( f(n) = z^n \) will have modulus \( r^n \) (decreasing) and argument \( n\theta \) (increasing). So the graph should be a spiral that approaches the origin as n increases, because the modulus gets smaller (since r < 1) and the argument increases, so it's a spiral. Wait, but the first graph is a spiral, the second is a circle (modulus constant), the third is a circle with modulus 1? Wait, no, the third graph has a point at (1,0), so modulus 1. But our modulus is less than 1, so as n increases, the modulus decreases, so the points get closer to the origin. So the first graph is a spiral (since modulus decreases, argument increases), the second is a circle (modulus constant), the third is a circle with modulus 1. So the correct graph should be the first one, the spiral. Wait, but let's check again. The complex number \( z = \frac{14}{15} + \frac{4}{15}i \), modulus \( r = \sqrt{(\frac{14}{15})^2 + (\frac{4}{15})^2} = \frac{\sqrt{196 + 16}}{15} = \frac{\sqrt{212}}{15} \approx 0.97 \), which is less than 1. So \( |z^n| = r^n \), which decreases as n increases (since r < 1). The argument of \( z^n \) is \( n\theta \), where \( \theta = \arg(z) = \arctan(\frac{4}{14}) = \arctan(\frac{2}{7}) \), so it's a constant angle per n, so the points rotate around the origin and get closer to it (since modulus decreases). So the graph should be a spiral, like the first graph (the left one) which shows a spiral towards the origin as n increases.

Step2: Analyze the graphs

- First graph: Spiral, modulus decreasing (since points get closer to origin), argument increasing (rotating).
- Second graph: Circle, modulus constant (since points are on a circle), so \( |z| = 1 \), but our \( |z| \approx 0.97 < 1 \), so not constant.
- Third graph: Circle with modulus 1 (point at (1,0)), but our modulus is less than 1, so not.

So the correct graph is the first one (the left graph).

Answer:

The graph on the Left