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

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Recursive functions are a fundamental concept in mathematics, allowing us to define a sequence of numbers by repeatedly applying a simple rule. In this article, we will explore a specific recursive function and use it to calculate the value of a sequence at a given position.

The Recursive Function

The recursive function we will be working with is defined as:

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 the sequence at position n+1n+1. The function is recursive because it is defined in terms of itself, with the value of the sequence at position n+1n+1 depending on the value of the sequence at position nn.

Initial Condition

We are given the initial condition:

f(1)=1f(1)=1

This tells us that the value of the sequence at position 1 is 1.

Calculating the Value of the Sequence

To calculate the value of the sequence at position 3, we need to apply the recursive function twice. First, we calculate the value of the sequence at position 2:

f(2)=−10f(1)=−10(1)=−10f(2)=-10 f(1)=-10(1)=-10

Next, we calculate the value of the sequence at position 3:

f(3)=−10f(2)=−10(−10)=100f(3)=-10 f(2)=-10(-10)=100

Therefore, the value of the sequence at position 3 is 100.

Conclusion

In this article, we have explored a recursive function and used it to calculate the value of a sequence at a given position. We have seen how the recursive function can be applied repeatedly to calculate the value of the sequence at different positions. This is a powerful tool in mathematics, allowing us to solve a wide range of problems.

Understanding the Options

Now that we have calculated the value of the sequence at position 3, let's take a look at the options:

A. 3 B. −30-30 C. 100 D. −1,000-1,000

We can see that option C, 100, is the correct answer.

Why is this Important?

Understanding recursive functions is important in mathematics because it allows us to solve a wide range of problems. Recursive functions can be used to model real-world phenomena, such as population growth and financial transactions. They can also be used to solve complex mathematical problems, such as finding the value of a sequence at a given position.

Real-World Applications

Recursive functions have many real-world applications. For example, they can be used to model population growth, where the population at a given time depends on the population at a previous time. They can also be used to model financial transactions, where the value of a transaction depends on the value of a previous transaction.

Conclusion

In conclusion, recursive functions are a powerful tool in mathematics, allowing us to solve a wide range of problems. They can be used to model real-world phenomena and solve complex mathematical problems. Understanding recursive functions is important for anyone interested in mathematics and its applications.

Final Thoughts

Recursive functions are a fundamental concept in mathematics, and understanding them is essential for anyone interested in mathematics and its applications. They can be used to model real-world phenomena and solve complex mathematical problems. In this article, we have explored a recursive function and used it to calculate the value of a sequence at a given position. We have seen how the recursive function can be applied repeatedly to calculate the value of the sequence at different positions. This is a powerful tool in mathematics, allowing us to solve a wide range of problems.

References

  • [1] "Recursive Functions" by Wikipedia
  • [2] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton
  • [3] "Discrete Mathematics and Its Applications" by Kenneth H. Rosen

Glossary

  • Recursive function: A function that is defined in terms of itself.
  • Sequence: A list of numbers that are related to each other by a rule.
  • Recursive: A function that is defined in terms of itself.

FAQs

  • Q: What is a recursive function? A: A recursive function is a function that is defined in terms of itself.
  • Q: What is a sequence? A: A sequence is a list of numbers that are related to each other by a rule.
  • Q: How do I calculate the value of a sequence at a given position? A: To calculate the value of a sequence at a given position, you need to apply the recursive function repeatedly until you reach the desired position.
    Recursive Functions: A Q&A Guide =====================================

In our previous article, we explored the concept of recursive functions and used them to calculate the value of a sequence at a given position. In this article, we will answer some frequently asked questions about recursive functions.

Q: What is a recursive function?

A: A recursive function is a function that is defined in terms of itself. This means that the function is called repeatedly with a smaller input until a base case is reached.

Q: What is a base case?

A: A base case is a value that is not defined in terms of the function itself. It is a value that is used to start the recursion.

Q: How do I know when to stop recursing?

A: You know when to stop recursing when you reach the base case. The base case is the value that is not defined in terms of the function itself.

Q: What is a recursive call?

A: A recursive call is a call to the function itself. This is what allows the function to be defined in terms of itself.

Q: What is the difference between a recursive function and a loop?

A: A recursive function and a loop are both used to repeat a process, but they do it in different ways. A recursive function calls itself repeatedly, while a loop uses a counter to repeat a process.

Q: When should I use a recursive function?

A: You should use a recursive function when the problem can be broken down into smaller sub-problems that are similar to the original problem. This is often the case with problems that have a recursive structure.

Q: What are some common examples of recursive functions?

A: Some common examples of recursive functions include:

  • Factorial: This is a function that calculates the factorial of a number.
  • Fibonacci sequence: This is a sequence of numbers where each number is the sum of the two preceding numbers.
  • Binary search: This is a function that searches for a value in a sorted list.

Q: How do I write a recursive function?

A: To write a recursive function, you need to define the function in terms of itself. This means that the function should call itself with a smaller input until a base case is reached.

Q: What are some common pitfalls to avoid when writing recursive functions?

A: Some common pitfalls to avoid when writing recursive functions include:

  • Infinite recursion: This is when the function calls itself repeatedly without reaching a base case.
  • Stack overflow: This is when the function calls itself too many times and the stack overflows.

Q: How do I debug a recursive function?

A: To debug a recursive function, you can use a debugger to step through the function and see where it is going wrong. You can also use print statements to see the values of the variables at each step.

Q: What are some real-world applications of recursive functions?

A: Some real-world applications of recursive functions include:

  • Compilers: These are programs that translate code from one language to another. They often use recursive functions to parse the code.
  • Database query optimizers: These are programs that optimize database queries. They often use recursive functions to find the optimal query plan.
  • Artificial intelligence: This is a field of computer science that deals with creating intelligent machines. Recursive functions are often used in artificial intelligence to model complex systems.

Conclusion

In this article, we have answered some frequently asked questions about recursive functions. We have seen how recursive functions can be used to solve complex problems and how they can be applied in real-world situations. We have also seen some common pitfalls to avoid when writing recursive functions and how to debug them.

Glossary

  • Recursive function: A function that is defined in terms of itself.
  • Base case: A value that is not defined in terms of the function itself.
  • Recursive call: A call to the function itself.
  • Loop: A way of repeating a process using a counter.
  • Infinite recursion: When the function calls itself repeatedly without reaching a base case.
  • Stack overflow: When the function calls itself too many times and the stack overflows.

FAQs

  • Q: What is a recursive function? A: A recursive function is a function that is defined in terms of itself.
  • Q: What is a base case? A: A base case is a value that is not defined in terms of the function itself.
  • Q: How do I know when to stop recursing? A: You know when to stop recursing when you reach the base case.
  • Q: What is a recursive call? A: A recursive call is a call to the function itself.
  • Q: What is the difference between a recursive function and a loop? A: A recursive function and a loop are both used to repeat a process, but they do it in different ways. A recursive function calls itself repeatedly, while a loop uses a counter to repeat a process.