Discover a world of knowledge at Westonci.ca, where experts and enthusiasts come together to answer your questions. Get immediate and reliable answers to your questions from a community of experienced professionals on our platform. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
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.
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.
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.