What Happens To $f(n)$ As $n$ Increases? F ( N ) = ∣ ( 11 12 + 7 12 I ) N ∣ F(n)=\left|\left(\frac{11}{12}+\frac{7}{12} I\right)^n\right| F ( N ) = ( 12 11 + 12 7 I ) N A. As $ N N N [/tex] Increases, $f(n)$ Remains Constant.B. As $n$ Increases,
What happens to $f(n)$ as $n$ increases?
Understanding the Function $f(n)$
The given function is $f(n)=\left|\left(\frac{11}{12}+\frac{7}{12} i\right)^n\right|$. This function involves complex numbers and absolute value. To understand what happens to $f(n)$ as $n$ increases, we need to break down the function and analyze its components.
Complex Numbers and Absolute Value
A complex number is a number that can be expressed in the form $a+bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, which satisfies $i^2=-1$. The absolute value of a complex number $z=a+bi$ is defined as $|z|=\sqrt{a2+b2}$.
In the given function, the complex number is $\frac{11}{12}+\frac{7}{12} i$. The absolute value of this complex number is $\left|\frac{11}{12}+\frac{7}{12} i\right|=\sqrt{\left(\frac{11}{12}\right)2+\left(\frac{7}{12}\right)2}=\sqrt{\frac{121}{144}+\frac{49}{144}}=\sqrt{\frac{170}{144}}=\frac{\sqrt{170}}{12}$.
Properties of Exponents
When dealing with exponents of complex numbers, we need to consider the properties of exponents. Specifically, we need to consider the property that states $|zn|=|z|n$, where $z$ is a complex number and $n$ is a positive integer.
Using this property, we can rewrite the function $f(n)$ as $f(n)=\left|\left(\frac{11}{12}+\frac{7}{12} i\right)^n\right|=\left|\frac{11}{12}+\frac{7}{12} i\right|n=\left(\frac{\sqrt{170}}{12}\right)n$.
Analyzing the Function
Now that we have rewritten the function $f(n)$, we can analyze its behavior as $n$ increases. Since the function is an exponential function with a base of $\frac{\sqrt{170}}{12}$, which is less than 1, we can expect the function to decrease as $n$ increases.
Conclusion
Based on our analysis, we can conclude that as $n$ increases, $f(n)$ decreases. This is because the function is an exponential function with a base of $\frac{\sqrt{170}}{12}$, which is less than 1.
Comparison with Options
Now that we have analyzed the function $f(n)$, we can compare our results with the options provided.
- Option A states that as $n$ increases, $f(n)$ remains constant. This is incorrect, as we have shown that $f(n)$ decreases as $n$ increases.
- Option B states that as $n$ increases, $f(n)$ decreases. This is correct, as we have shown that $f(n)$ decreases as $n$ increases.
Final Answer
Based on our analysis, we can conclude that the correct answer is:
- B. As $n$ increases, $f(n)$ decreases.
Additional Discussion
In this discussion, we have analyzed the function $f(n)=\left|\left(\frac{11}{12}+\frac{7}{12} i\right)^n\right|$. We have shown that as $n$ increases, $f(n)$ decreases. This is because the function is an exponential function with a base of $\frac{\sqrt{170}}{12}$, which is less than 1.
Mathematical Background
The mathematical background for this discussion is complex numbers and absolute value. We have used the property that states $|zn|=|z|n$, where $z$ is a complex number and $n$ is a positive integer.
Real-World Applications
The real-world applications of this discussion are limited. However, the mathematical concepts used in this discussion are fundamental to many areas of mathematics, including algebra and analysis.
Future Research
Future research in this area could involve exploring the properties of complex numbers and absolute value in more detail. This could involve analyzing the behavior of complex numbers and absolute value in different mathematical contexts, such as algebra and analysis.
Conclusion
In conclusion, we have analyzed the function $f(n)=\left|\left(\frac{11}{12}+\frac{7}{12} i\right)^n\right|$. We have shown that as $n$ increases, $f(n)$ decreases. This is because the function is an exponential function with a base of $\frac{\sqrt{170}}{12}$, which is less than 1.
Q&A: What happens to $f(n)$ as $n$ increases?
Q: What is the function $f(n)$?
A: The function $f(n)$ is defined as $f(n)=\left|\left(\frac{11}{12}+\frac{7}{12} i\right)^n\right|$. This function involves complex numbers and absolute value.
Q: What is the absolute value of a complex number?
A: The absolute value of a complex number $z=a+bi$ is defined as $|z|=\sqrt{a2+b2}$.
Q: How does the property $|zn|=|z|n$ relate to the function $f(n)$?
A: The property $|zn|=|z|n$ states that the absolute value of a complex number raised to a power is equal to the absolute value of the complex number raised to that power. This property is used to rewrite the function $f(n)$ as $f(n)=\left|\left(\frac{11}{12}+\frac{7}{12} i\right)^n\right|=\left|\frac{11}{12}+\frac{7}{12} i\right|n=\left(\frac{\sqrt{170}}{12}\right)n$.
Q: What happens to $f(n)$ as $n$ increases?
A: As $n$ increases, $f(n)$ decreases. This is because the function is an exponential function with a base of $\frac{\sqrt{170}}{12}$, which is less than 1.
Q: Why does $f(n)$ decrease as $n$ increases?
A: $f(n)$ decreases as $n$ increases because the function is an exponential function with a base of $\frac{\sqrt{170}}{12}$, which is less than 1. This means that as $n$ increases, the value of $f(n)$ will decrease.
Q: What is the correct answer?
A: The correct answer is:
- B. As $n$ increases, $f(n)$ decreases.
Q: What are some real-world applications of this discussion?
A: The real-world applications of this discussion are limited. However, the mathematical concepts used in this discussion are fundamental to many areas of mathematics, including algebra and analysis.
Q: What are some potential areas for future research?
A: Potential areas for future research include exploring the properties of complex numbers and absolute value in more detail. This could involve analyzing the behavior of complex numbers and absolute value in different mathematical contexts, such as algebra and analysis.
Q: What is the significance of the function $f(n)$?
A: The function $f(n)$ is significant because it provides a way to analyze the behavior of complex numbers and absolute value. This can be useful in a variety of mathematical contexts, including algebra and analysis.
Q: How does the function $f(n)$ relate to other mathematical concepts?
A: The function $f(n)$ relates to other mathematical concepts, such as exponential functions and complex numbers. Understanding the behavior of $f(n)$ can provide insight into the behavior of these other concepts.
Q: What are some potential uses of the function $f(n)$?
A: Potential uses of the function $f(n)$ include analyzing the behavior of complex numbers and absolute value in different mathematical contexts. This can be useful in a variety of fields, including mathematics, physics, and engineering.
Q: How does the function $f(n)$ compare to other functions?
A: The function $f(n)$ compares to other functions in that it is an exponential function with a base of $\frac{\sqrt{170}}{12}$, which is less than 1. This means that as $n$ increases, the value of $f(n)$ will decrease.
Q: What are some potential limitations of the function $f(n)$?
A: Potential limitations of the function $f(n)$ include the fact that it is only defined for positive integer values of $n$. Additionally, the function may not be well-defined for other types of numbers, such as negative integers or non-integer values.
Q: How does the function $f(n)$ relate to the concept of convergence?
A: The function $f(n)$ relates to the concept of convergence in that it can be used to analyze the behavior of complex numbers and absolute value as $n$ increases. This can provide insight into whether or not a sequence of complex numbers converges to a particular value.
Q: What are some potential applications of the function $f(n)$ in other fields?
A: Potential applications of the function $f(n)$ in other fields include physics, engineering, and computer science. In these fields, the function can be used to analyze the behavior of complex systems and make predictions about their behavior.
Q: How does the function $f(n)$ compare to other mathematical functions?
A: The function $f(n)$ compares to other mathematical functions in that it is an exponential function with a base of $\frac{\sqrt{170}}{12}$, which is less than 1. This means that as $n$ increases, the value of $f(n)$ will decrease.
Q: What are some potential uses of the function $f(n)$ in mathematical modeling?
A: Potential uses of the function $f(n)$ in mathematical modeling include analyzing the behavior of complex systems and making predictions about their behavior. This can be useful in a variety of fields, including physics, engineering, and computer science.
Q: How does the function $f(n)$ relate to the concept of asymptotic behavior?
A: The function $f(n)$ relates to the concept of asymptotic behavior in that it can be used to analyze the behavior of complex numbers and absolute value as $n$ increases. This can provide insight into whether or not a sequence of complex numbers converges to a particular value.
Q: What are some potential applications of the function $f(n)$ in data analysis?
A: Potential applications of the function $f(n)$ in data analysis include analyzing the behavior of complex systems and making predictions about their behavior. This can be useful in a variety of fields, including physics, engineering, and computer science.
Q: How does the function $f(n)$ compare to other mathematical functions in terms of its rate of convergence?
A: The function $f(n)$ compares to other mathematical functions in terms of its rate of convergence in that it is an exponential function with a base of $\frac{\sqrt{170}}{12}$, which is less than 1. This means that as $n$ increases, the value of $f(n)$ will decrease at a rate that is faster than many other mathematical functions.