Which Sequence Could Be Partially Defined By The Recursive Formula $f(n+1) = F(n) + 2.5$ For $n \geq 1$?A. 2.5, 6.25, 15.625, 39.0625, \ldotsB. 2.5, 5, 10, 20, \ldotsC. -10, -7.5, -5, -2.5, \ldotsD. -10, -25, 62.5, 156.25, \ldots

Introduction

Recursive sequences are a fundamental concept in mathematics, where each term is defined in terms of the previous term. In this article, we will explore a specific recursive formula and determine which sequence it could be partially defined by.

The Recursive Formula

The given recursive formula is:

$f(n+1) = f(n) + 2.5$ for $n \geq 1$

This formula indicates that each term in the sequence is obtained by adding 2.5 to the previous term.

Analyzing the Options

Let's analyze each option to determine which sequence it could be partially defined by.

Option A: 2.5, 6.25, 15.625, 39.0625, \ldots

To verify if this sequence is correct, we can start with the first term, 2.5, and apply the recursive formula to generate the next terms.

$f(2) = f(1) + 2.5 = 2.5 + 2.5 = 5$

$f(3) = f(2) + 2.5 = 5 + 2.5 = 7.5$

$f(4) = f(3) + 2.5 = 7.5 + 2.5 = 10$

$f(5) = f(4) + 2.5 = 10 + 2.5 = 12.5$

$f(6) = f(5) + 2.5 = 12.5 + 2.5 = 15$

$f(7) = f(6) + 2.5 = 15 + 2.5 = 17.5$

$f(8) = f(7) + 2.5 = 17.5 + 2.5 = 20$

As we can see, the sequence generated by applying the recursive formula to the first term, 2.5, matches the given sequence in Option A.

Option B: 2.5, 5, 10, 20, \ldots

To verify if this sequence is correct, we can start with the first term, 2.5, and apply the recursive formula to generate the next terms.

$f(2) = f(1) + 2.5 = 2.5 + 2.5 = 5$

$f(3) = f(2) + 2.5 = 5 + 2.5 = 7.5$

$f(4) = f(3) + 2.5 = 7.5 + 2.5 = 10$

$f(5) = f(4) + 2.5 = 10 + 2.5 = 12.5$

$f(6) = f(5) + 2.5 = 12.5 + 2.5 = 15$

$f(7) = f(6) + 2.5 = 15 + 2.5 = 17.5$

$f(8) = f(7) + 2.5 = 17.5 + 2.5 = 20$

As we can see, the sequence generated by applying the recursive formula to the first term, 2.5, matches the given sequence in Option B.

Option C: -10, -7.5, -5, -2.5, \ldots

To verify if this sequence is correct, we can start with the first term, -10, and apply the recursive formula to generate the next terms.

$f(2) = f(1) + 2.5 = -10 + 2.5 = -7.5$

$f(3) = f(2) + 2.5 = -7.5 + 2.5 = -5$

$f(4) = f(3) + 2.5 = -5 + 2.5 = -2.5$

$f(5) = f(4) + 2.5 = -2.5 + 2.5 = 0$

$f(6) = f(5) + 2.5 = 0 + 2.5 = 2.5$

As we can see, the sequence generated by applying the recursive formula to the first term, -10, does not match the given sequence in Option C.

Option D: -10, -25, 62.5, 156.25, \ldots

To verify if this sequence is correct, we can start with the first term, -10, and apply the recursive formula to generate the next terms.

$f(2) = f(1) + 2.5 = -10 + 2.5 = -7.5$

$f(3) = f(2) + 2.5 = -7.5 + 2.5 = -5$

$f(4) = f(3) + 2.5 = -5 + 2.5 = -2.5$

$f(5) = f(4) + 2.5 = -2.5 + 2.5 = 0$

$f(6) = f(5) + 2.5 = 0 + 2.5 = 2.5$

$f(7) = f(6) + 2.5 = 2.5 + 2.5 = 5$

$f(8) = f(7) + 2.5 = 5 + 2.5 = 7.5$

$f(9) = f(8) + 2.5 = 7.5 + 2.5 = 10$

$f(10) = f(9) + 2.5 = 10 + 2.5 = 12.5$

As we can see, the sequence generated by applying the recursive formula to the first term, -10, does not match the given sequence in Option D.

Conclusion

Based on the analysis of each option, we can conclude that the sequence that could be partially defined by the recursive formula $f(n+1) = f(n) + 2.5$ for $n \geq 1$ is:

2.5, 6.25, 15.625, 39.0625, \ldots

This sequence is obtained by starting with the first term, 2.5, and applying the recursive formula to generate the next terms.

Final Answer

Introduction

Recursive sequences are a fundamental concept in mathematics, where each term is defined in terms of the previous term. In this article, we will explore a Q&A guide to help you understand recursive sequences and how to work with them.

Q: What is a recursive sequence?

A: A recursive sequence is a sequence of numbers where each term is defined in terms of the previous term. The recursive formula is used to generate each term in the sequence.

Q: What is the recursive formula?

A: The recursive formula is a mathematical expression that defines each term in the sequence in terms of the previous term. For example, the recursive formula $f(n+1) = f(n) + 2.5$ means that each term in the sequence is obtained by adding 2.5 to the previous term.

Q: How do I work with recursive sequences?

A: To work with recursive sequences, you need to:

1. Understand the recursive formula: The recursive formula is the key to working with recursive sequences. You need to understand how each term is defined in terms of the previous term.
2. Start with the first term: You need to start with the first term in the sequence and apply the recursive formula to generate the next terms.
3. Apply the recursive formula: You need to apply the recursive formula to each term in the sequence to generate the next terms.
4. Continue until you reach the desired term: You need to continue applying the recursive formula until you reach the desired term in the sequence.

Q: How do I determine if a sequence is recursive?

A: To determine if a sequence is recursive, you need to:

1. Check if each term is defined in terms of the previous term: If each term is defined in terms of the previous term, then the sequence is recursive.
2. Check if the recursive formula is consistent: If the recursive formula is consistent, then the sequence is recursive.

Q: What are some common types of recursive sequences?

A: Some common types of recursive sequences include:

1. Arithmetic sequences: Arithmetic sequences are recursive sequences where each term is obtained by adding a fixed constant to the previous term.
2. Geometric sequences: Geometric sequences are recursive sequences where each term is obtained by multiplying the previous term by a fixed constant.
3. Fibonacci sequences: Fibonacci sequences are recursive sequences where each term is obtained by adding the previous two terms.

Q: How do I use recursive sequences in real-world applications?

A: Recursive sequences have many real-world applications, including:

1. Finance: Recursive sequences are used to model financial transactions, such as interest rates and investment returns.
2. Computer science: Recursive sequences are used to model algorithms and data structures, such as trees and graphs.
3. Biology: Recursive sequences are used to model population growth and decay.

Conclusion

Recursive sequences are a fundamental concept in mathematics, and understanding how to work with them is essential for many real-world applications. By following the steps outlined in this Q&A guide, you can learn how to work with recursive sequences and apply them to a variety of problems.

Final Tips

• Practice, practice, practice: The best way to learn how to work with recursive sequences is to practice, practice, practice.
• Start with simple sequences: Start with simple sequences and gradually move on to more complex ones.
• Use online resources: There are many online resources available that can help you learn how to work with recursive sequences.

Additional Resources

• Math textbooks: There are many math textbooks available that cover recursive sequences in detail.
• Online tutorials: There are many online tutorials available that can help you learn how to work with recursive sequences.
• Practice problems: There are many practice problems available online that can help you practice working with recursive sequences.