A Sequence Is Defined By The Recursive Function $f(n+1) = -10 F(n$\].If $f(1) = 1$, What Is $f(3$\]?A. \[$-30\$\]B. \[$-1,000\$\]C. 100D. 3

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Introduction

In mathematics, recursive sequences are a fundamental concept that helps us understand the behavior of functions and their values at different points. In this article, we will delve into the world of recursive sequences and explore the function f(n+1)=−10f(n)f(n+1) = -10 f(n), where f(1)=1f(1) = 1. Our goal is to find the value of f(3)f(3).

Understanding Recursive Sequences

A recursive sequence is a sequence of numbers that is defined recursively, meaning that each term is defined in terms of previous terms. In the case of the function f(n+1)=−10f(n)f(n+1) = -10 f(n), each term is defined as −10-10 times the previous term.

The Recursive Function f(n+1)=−10f(n)f(n+1) = -10 f(n)

Let's start by understanding the recursive function f(n+1)=−10f(n)f(n+1) = -10 f(n). This function takes a positive integer nn as input and returns the value of f(n+1)f(n+1), which is defined as −10-10 times the value of f(n)f(n).

Calculating f(2)f(2)

To calculate f(2)f(2), we need to use the recursive function f(n+1)=−10f(n)f(n+1) = -10 f(n). Since f(1)=1f(1) = 1, we can plug in n=1n = 1 into the function to get:

f(2)=−10f(1)=−10⋅1=−10f(2) = -10 f(1) = -10 \cdot 1 = -10

Calculating f(3)f(3)

Now that we have the value of f(2)f(2), we can use it to calculate f(3)f(3). We can plug in n=2n = 2 into the recursive function to get:

f(3)=−10f(2)=−10⋅(−10)=100f(3) = -10 f(2) = -10 \cdot (-10) = 100

Conclusion

In this article, we have explored the recursive sequence defined by the function f(n+1)=−10f(n)f(n+1) = -10 f(n), where f(1)=1f(1) = 1. We have calculated the values of f(2)f(2) and f(3)f(3) using the recursive function and found that f(3)=100f(3) = 100.

Final Answer

The final answer is 100\boxed{100}.

Additional Information

  • The recursive function f(n+1)=−10f(n)f(n+1) = -10 f(n) is an example of a linear recursive sequence.
  • The value of f(n)f(n) can be calculated using the recursive function f(n+1)=−10f(n)f(n+1) = -10 f(n).
  • The sequence f(n)f(n) is an example of a sequence that converges to 00 as nn approaches infinity.

References

  • [1] "Recursive Sequences" by MathWorld
  • [2] "Linear Recursive Sequences" by Wolfram MathWorld

Related Topics

  • Recursive sequences
  • Linear recursive sequences
  • Convergence of sequences

Frequently Asked Questions

  • Q: What is a recursive sequence?
  • A: A recursive sequence is a sequence of numbers that is defined recursively, meaning that each term is defined in terms of previous terms.
  • Q: How do I calculate the value of f(n)f(n) using the recursive function f(n+1)=−10f(n)f(n+1) = -10 f(n)?
  • A: To calculate the value of f(n)f(n), you can plug in nn into the recursive function and use the value of f(n−1)f(n-1) to calculate the value of f(n)f(n).

Glossary

  • Recursive sequence: A sequence of numbers that is defined recursively, meaning that each term is defined in terms of previous terms.
  • Linear recursive sequence: A recursive sequence where each term is defined as a linear combination of previous terms.
  • Convergence of sequences: The property of a sequence that approaches a limit as the index of the sequence approaches infinity.
    A Recursive Sequence: Unraveling the Mystery of f(n)f(n) ===========================================================

Q&A: Recursive Sequences and Beyond

Q: What is a recursive sequence?

A: A recursive sequence is a sequence of numbers that is defined recursively, meaning that each term is defined in terms of previous terms. In the case of the function f(n+1)=−10f(n)f(n+1) = -10 f(n), each term is defined as −10-10 times the previous term.

Q: How do I calculate the value of f(n)f(n) using the recursive function f(n+1)=−10f(n)f(n+1) = -10 f(n)?

A: To calculate the value of f(n)f(n), you can plug in nn into the recursive function and use the value of f(n−1)f(n-1) to calculate the value of f(n)f(n). For example, to calculate f(3)f(3), you can use the recursive function as follows:

f(3)=−10f(2)=−10⋅(−10)=100f(3) = -10 f(2) = -10 \cdot (-10) = 100

Q: What is the difference between a recursive sequence and an iterative sequence?

A: A recursive sequence is a sequence that is defined recursively, meaning that each term is defined in terms of previous terms. An iterative sequence, on the other hand, is a sequence that is defined iteratively, meaning that each term is defined in terms of the previous term and a fixed set of operations.

Q: Can you give an example of an iterative sequence?

A: Yes, the sequence f(n)=2f(n−1)+1f(n) = 2 f(n-1) + 1 is an example of an iterative sequence. To calculate the value of f(n)f(n), you can use the iterative formula as follows:

f(1)=1f(1) = 1 f(2)=2f(1)+1=3f(2) = 2 f(1) + 1 = 3 f(3)=2f(2)+1=7f(3) = 2 f(2) + 1 = 7

Q: What is the relationship between recursive sequences and linear algebra?

A: Recursive sequences and linear algebra are closely related. In fact, many recursive sequences can be represented as linear systems of equations. For example, the recursive sequence f(n+1)=−10f(n)f(n+1) = -10 f(n) can be represented as a linear system of equations as follows:

f(n+1)=−10f(n)f(n+1) = -10 f(n) f(1)=1f(1) = 1

Q: Can you give an example of a recursive sequence that can be represented as a linear system of equations?

A: Yes, the recursive sequence f(n+1)=2f(n)+1f(n+1) = 2 f(n) + 1 can be represented as a linear system of equations as follows:

f(n+1)=2f(n)+1f(n+1) = 2 f(n) + 1 f(1)=1f(1) = 1

Q: What is the significance of recursive sequences in computer science?

A: Recursive sequences are widely used in computer science to model real-world phenomena, such as population growth, financial markets, and network traffic. They are also used in algorithms and data structures, such as recursive sorting and searching.

Q: Can you give an example of a recursive sequence in computer science?

A: Yes, the recursive sequence f(n)=2f(n−1)+1f(n) = 2 f(n-1) + 1 is used in computer science to model the growth of a population. The sequence represents the number of individuals in the population at each time step.

Q: What is the relationship between recursive sequences and dynamical systems?

A: Recursive sequences and dynamical systems are closely related. In fact, many recursive sequences can be represented as dynamical systems. For example, the recursive sequence f(n+1)=−10f(n)f(n+1) = -10 f(n) can be represented as a dynamical system as follows:

xn+1=−10xnx_{n+1} = -10 x_n x1=1x_1 = 1

Q: Can you give an example of a recursive sequence that can be represented as a dynamical system?

A: Yes, the recursive sequence f(n+1)=2f(n)+1f(n+1) = 2 f(n) + 1 can be represented as a dynamical system as follows:

xn+1=2xn+1x_{n+1} = 2 x_n + 1 x1=1x_1 = 1

Glossary

  • Recursive sequence: A sequence of numbers that is defined recursively, meaning that each term is defined in terms of previous terms.
  • Iterative sequence: A sequence that is defined iteratively, meaning that each term is defined in terms of the previous term and a fixed set of operations.
  • Linear algebra: A branch of mathematics that deals with the study of linear equations and linear transformations.
  • Dynamical system: A system that is described by a set of differential equations or difference equations that describe the evolution of the system over time.

References

  • [1] "Recursive Sequences" by MathWorld
  • [2] "Linear Algebra" by Wolfram MathWorld
  • [3] "Dynamical Systems" by Wolfram MathWorld

Related Topics

  • Recursive sequences
  • Iterative sequences
  • Linear algebra
  • Dynamical systems

Frequently Asked Questions

  • Q: What is a recursive sequence?
  • A: A recursive sequence is a sequence of numbers that is defined recursively, meaning that each term is defined in terms of previous terms.
  • Q: How do I calculate the value of f(n)f(n) using the recursive function f(n+1)=−10f(n)f(n+1) = -10 f(n)?
  • A: To calculate the value of f(n)f(n), you can plug in nn into the recursive function and use the value of f(n−1)f(n-1) to calculate the value of f(n)f(n).