To analyze the behavior of the function [tex]\( f(n) = \left|(0.5 + 0.2i)^n\right| \)[/tex] as [tex]\( n \)[/tex] increases, we need to consider the properties of complex numbers and their magnitudes.
1. Understanding the Magnitude of a Complex Number:
Given a complex number [tex]\( z = a + bi \)[/tex], its magnitude is defined as [tex]\( |z| = \sqrt{a^2 + b^2} \)[/tex].
2. Determine the Magnitude of the Base Complex Number:
For the complex number [tex]\( 0.5 + 0.2i \)[/tex], calculate its magnitude:
[tex]\[
|0.5 + 0.2i| = \sqrt{(0.5)^2 + (0.2)^2} = \sqrt{0.25 + 0.04} = \sqrt{0.29} \approx 0.5385.
\][/tex]
3. Analyzing the Magnitude Over [tex]\( n \)[/tex]:
When you raise the magnitude of a number to the power of [tex]\( n \)[/tex], the overall magnitude changes accordingly. If the magnitude of the base complex number [tex]\( |0.5 + 0.2i| \approx 0.5385 \)[/tex] is less than 1, raising it to higher powers (i.e., increasing [tex]\( n \)[/tex]) will make it smaller:
[tex]\[
|(0.5 + 0.2i)^n| = (|0.5 + 0.2i|)^n.
\][/tex]
4. Behavior of [tex]\( f(n) \)[/tex]:
Since [tex]\( |0.5 + 0.2i| \approx 0.5385 \)[/tex] is less than 1, raising this magnitude to higher and higher powers [tex]\( n \)[/tex] will cause the result to approach zero:
[tex]\[
(0.5385)^n \rightarrow 0 \quad \text{as} \quad n \rightarrow \infty.
\][/tex]
Based on this analysis, we can conclude that as [tex]\( n \)[/tex] increases, [tex]\( f(n) = \left|(0.5 + 0.2i)^n\right| \)[/tex] decreases.
Thus, the correct statement is:
- As [tex]\( n \)[/tex] increases, [tex]\( f(n) \)[/tex] decreases.
