Question

find the graph of this function as the value of n increases, starting from n = 1. f(n) = (4/7 + 4/5 i)^n remember |a + bi| = √(a² + b²)

Explanation:

Step1: Find the modulus of the complex number

The complex number is \( z = \frac{4}{7} + \frac{4}{5}i \). The modulus \( r \) of a complex number \( a + bi \) is \( r=\sqrt{a^{2}+b^{2}} \). So, \( r = \sqrt{(\frac{4}{7})^{2}+(\frac{4}{5})^{2}}=\sqrt{\frac{16}{49}+\frac{16}{25}}=\sqrt{\frac{16\times25 + 16\times49}{49\times25}}=\sqrt{\frac{16(25 + 49)}{1225}}=\sqrt{\frac{16\times74}{1225}}=\frac{4\sqrt{74}}{35}\approx\frac{4\times8.602}{35}\approx\frac{34.408}{35}\approx0.983 \). Since \( r\approx0.983< 1 \), as \( n \) increases, the modulus of \( z^{n} \) is \( r^{n} \), which will decrease (because \( r < 1 \) and when we raise it to higher powers, it gets smaller).

Step2: Analyze the behavior of the graph

For a complex number \( z = r(\cos\theta+i\sin\theta) \), \( z^{n}=r^{n}(\cos(n\theta)+i\sin(n\theta)) \). The modulus \( r^{n} \) decreases as \( n \) increases (since \( r<1 \)), so the points \( z^{n} \) should get closer to the origin as \( n \) increases. Looking at the three graphs, the first graph (left - most) has points that seem to have a modulus decreasing (since the circle is around radius less than 1 and the points are arranged such that as \( n \) increases, they move towards the origin? Wait, no, actually, when \( r < 1 \), \( r^{n}\to0 \) as \( n\to\infty \), so the distance from the origin should decrease. The first graph (left) has a circle with radius less than 1 (since the unit circle would have radius 1, and this is inside? Wait, no, the modulus we calculated is less than 1, so the initial point (n = 1) is at \( r\approx0.983 \), and as \( n \) increases, \( r^{n} \) decreases. So the points should lie on a spiral towards the origin? Wait, no, actually, for each \( n \), the modulus is \( r^{n} \) and the argument is \( n\theta \). So the set of points \( z^{n} \) will lie on a spiral (Archimedean spiral? No, it's a geometric spiral) where the modulus decreases by a factor of \( r \) each time and the argument increases by \( \theta \). But among the three graphs, the left - most graph has points that seem to be getting closer to the origin as \( n \) increases (since the radius of the "circle" of points is decreasing), the middle graph has a constant radius (so \( r = 1 \), which is not our case), and the right - most graph has a radius increasing (so \( r>1 \), which is not our case). Wait, maybe I made a mistake in modulus calculation. Wait, \( (\frac{4}{7})\approx0.571 \), \( (\frac{4}{5}) = 0.8 \). Then \( (\frac{4}{7})^{2}=0.326 \), \( (\frac{4}{5})^{2}=0.64 \), sum is \( 0.326 + 0.64 = 0.966 \), square root is \( \sqrt{0.966}\approx0.983 \), which is less than 1. So as \( n \) increases, \( r^{n} \) decreases. So the points \( z^{n} \) should be on a spiral where the distance from the origin decreases. The left - most graph shows points that are arranged such that as \( n \) increases (starting from \( n = 1 \)), the distance from the origin decreases (since the modulus \( r^{n} \) decreases). The middle graph has points on a circle of constant radius (so \( r = 1 \)), and the right - most graph has points on a circle of increasing radius (so \( r>1 \)). So the correct graph is the left - most one.

Answer:

The Left Graph