At Westonci.ca, we connect you with the answers you need, thanks to our active and informed community. Join our Q&A platform and get accurate answers to all your questions from professionals across multiple disciplines. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields 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.
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.