Find the graph of this function as the value of n increases, starting from [tex]n =1[/tex].
[tex]f(n)=\left(\frac{1}{2}+\frac{4}{5} i\right)^{n}[/tex]
Remember: [tex]| a + b i|=\sqrt{ a ^2+ b ^2}[/tex]
Answer (2)
As n increases in the function f(n) = (1/2 + 4/5 i)^{n}, the graph spirals inward towards the origin due to the magnitude being less than 1. The points rotate around the origin, converging closer to it with each increase in n. Overall, this results in a spiral pattern reflecting both declining magnitude and changing angle.
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Convert the complex number to polar form: z = r e i θ .
Calculate the magnitude: r = 10 89 ≈ 0.943 < 1 .
Calculate the argument: θ = arctan ( 5 8 ) ≈ 1.012 radians.
The graph of f ( n ) is a spiral converging to the origin as n increases, with decreasing magnitude and rotating argument. Spiral converging to the origin
Explanation
Understanding the Problem We are given a function f ( n ) = ( 2 1 + 5 4 i ) n , where n is an integer starting from 1. We want to describe the behavior of the graph of this function as n increases. The magnitude of a complex number a + bi is given by ∣ a + bi ∣ = a 2 + b 2 .
Converting to Polar Form First, let's convert the complex number z = 2 1 + 5 4 i to polar form r e i θ , where r = ∣ z ∣ and θ = arctan ( Re ( z ) Im ( z ) ) .
Calculating the Magnitude Calculate the magnitude r = 2 1 + 5 4 i = ( 2 1 ) 2 + ( 5 4 ) 2 = 4 1 + 25 16 = 100 25 + 64 = 100 89 = 10 89 ≈ 0.943 .
Calculating the Argument Calculate the argument θ = arctan ( 1/2 4/5 ) = arctan ( 5 8 ) ≈ 1.012 radians.
Expressing f(n) in Polar Form Express f ( n ) as f ( n ) = ( r e i θ ) n = r n e in θ = r n ( cos ( n θ ) + i sin ( n θ )) .
Analyzing the Magnitude Since r = 10 89 < 1 , r n approaches 0 as n increases. Therefore, the magnitude of f ( n ) decreases as n increases.
Analyzing the Argument The argument of f ( n ) is n θ = n arctan ( 5 8 ) . As n increases, the points rotate around the origin.
Describing the Graph The graph of f ( n ) will be a spiral that converges to the origin as n increases. Each point is closer to the origin than the previous one, and rotated by an angle of approximately 1.012 radians.
Examples
Understanding the behavior of complex functions like f ( n ) is crucial in fields such as signal processing and quantum mechanics. In signal processing, the function's spiral trajectory can represent the damping of a signal over time, where the magnitude decreases and the phase changes. In quantum mechanics, similar functions describe the evolution of quantum states, where the magnitude relates to probability and the phase to the state's oscillation. Visualizing these functions helps engineers and physicists design filters, analyze quantum systems, and predict system behavior.
