Which Recursive Formula Can Be Used To Generate The Sequence Shown, Where $f(1) = 9.6$ And N ≥ 1 N \geq 1 N ≥ 1 ?Sequence: 9.6 , − 4.8 , 2.4 , − 1.2 , 0.6 , … 9.6, -4.8, 2.4, -1.2, 0.6, \ldots 9.6 , − 4.8 , 2.4 , − 1.2 , 0.6 , … A. F ( N + 1 ) = ( − 0.5 ) F ( N F(n+1) = (-0.5) F(n F ( N + 1 ) = ( − 0.5 ) F ( N ]B. F ( N + 1 ) = ( 0.5 ) F ( N F(n+1) = (0.5) F(n F ( N + 1 ) = ( 0.5 ) F ( N ]C.

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Introduction

In mathematics, a recursive formula is a way to define a sequence where each term is defined in terms of previous terms. This type of formula is essential in various mathematical and computational applications. In this article, we will explore a recursive formula for a given sequence and determine which formula can be used to generate the sequence.

The Sequence

The given sequence is: 9.6,−4.8,2.4,−1.2,0.6,…9.6, -4.8, 2.4, -1.2, 0.6, \ldots We are also given that f(1)=9.6f(1) = 9.6 and n≥1n \geq 1. Our goal is to find the recursive formula that can be used to generate this sequence.

Recursive Formula Options

There are three options for the recursive formula:

A. f(n+1)=(−0.5)f(n)f(n+1) = (-0.5) f(n) B. f(n+1)=(0.5)f(n)f(n+1) = (0.5) f(n) C. Other (not specified)

Analyzing Option A

Let's start by analyzing option A: f(n+1)=(−0.5)f(n)f(n+1) = (-0.5) f(n). To determine if this formula can generate the given sequence, we can start by calculating the first few terms.

f(2)=(−0.5)f(1)=(−0.5)(9.6)=−4.8f(2) = (-0.5) f(1) = (-0.5)(9.6) = -4.8

f(3)=(−0.5)f(2)=(−0.5)(−4.8)=2.4f(3) = (-0.5) f(2) = (-0.5)(-4.8) = 2.4

f(4)=(−0.5)f(3)=(−0.5)(2.4)=−1.2f(4) = (-0.5) f(3) = (-0.5)(2.4) = -1.2

f(5)=(−0.5)f(4)=(−0.5)(−1.2)=0.6f(5) = (-0.5) f(4) = (-0.5)(-1.2) = 0.6

As we can see, the first few terms of the sequence generated by option A match the given sequence. This suggests that option A may be the correct recursive formula.

Analyzing Option B

Now, let's analyze option B: f(n+1)=(0.5)f(n)f(n+1) = (0.5) f(n). To determine if this formula can generate the given sequence, we can start by calculating the first few terms.

f(2)=(0.5)f(1)=(0.5)(9.6)=4.8f(2) = (0.5) f(1) = (0.5)(9.6) = 4.8

f(3)=(0.5)f(2)=(0.5)(4.8)=2.4f(3) = (0.5) f(2) = (0.5)(4.8) = 2.4

f(4)=(0.5)f(3)=(0.5)(2.4)=1.2f(4) = (0.5) f(3) = (0.5)(2.4) = 1.2

f(5)=(0.5)f(4)=(0.5)(1.2)=0.6f(5) = (0.5) f(4) = (0.5)(1.2) = 0.6

As we can see, the first few terms of the sequence generated by option B do not match the given sequence. This suggests that option B is not the correct recursive formula.

Conclusion

Based on our analysis, we can conclude that the recursive formula that can be used to generate the given sequence is:

f(n+1)=(−0.5)f(n)f(n+1) = (-0.5) f(n)

This formula matches the given sequence and is consistent with the initial condition f(1)=9.6f(1) = 9.6.

Discussion

The recursive formula f(n+1)=(−0.5)f(n)f(n+1) = (-0.5) f(n) is a simple and elegant way to generate the given sequence. This formula can be used in various mathematical and computational applications, such as modeling population growth or decay, or generating random numbers.

In conclusion, the recursive formula that can be used to generate the given sequence is f(n+1)=(−0.5)f(n)f(n+1) = (-0.5) f(n). This formula is consistent with the initial condition and matches the given sequence.

References

  • [1] "Recursive Sequences" by Math Open Reference
  • [2] "Recursive Formulas" by Wolfram MathWorld

Additional Resources

  • [1] "Recursive Sequences" by Khan Academy
  • [2] "Recursive Formulas" by MIT OpenCourseWare

Glossary

  • Recursive Formula: A formula that defines a sequence where each term is defined in terms of previous terms.
  • Initial Condition: The value of the first term in a sequence.
  • Sequence: A list of numbers or values that follow a specific pattern or rule.

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Introduction

In our previous article, we explored the recursive formula for a given sequence and determined that the correct formula is f(n+1)=(−0.5)f(n)f(n+1) = (-0.5) f(n). In this article, we will answer some frequently asked questions about recursive formulas and sequences.

Q&A

Q: What is a recursive formula?

A: A recursive formula is a way to define a sequence where each term is defined in terms of previous terms.

Q: How do I determine the correct recursive formula for a given sequence?

A: To determine the correct recursive formula, you need to analyze the sequence and look for a pattern or rule that defines each term in terms of previous terms.

Q: What is the initial condition in a recursive formula?

A: The initial condition is the value of the first term in a sequence.

Q: Can a recursive formula have multiple initial conditions?

A: No, a recursive formula typically has only one initial condition.

Q: How do I calculate the next term in a sequence using a recursive formula?

A: To calculate the next term in a sequence using a recursive formula, you need to substitute the previous term into the formula and simplify.

Q: Can a recursive formula be used to generate a sequence with negative terms?

A: Yes, a recursive formula can be used to generate a sequence with negative terms.

Q: Can a recursive formula be used to generate a sequence with fractional terms?

A: Yes, a recursive formula can be used to generate a sequence with fractional terms.

Q: How do I determine if a recursive formula is correct?

A: To determine if a recursive formula is correct, you need to check if it matches the given sequence and if it is consistent with the initial condition.

Q: Can a recursive formula be used to model real-world phenomena?

A: Yes, a recursive formula can be used to model real-world phenomena such as population growth or decay, or financial transactions.

Q: Can a recursive formula be used to generate random numbers?

A: Yes, a recursive formula can be used to generate random numbers.

Examples

Example 1: Recursive Formula for a Sequence with Positive Terms

Suppose we have a sequence with positive terms: 2,4,8,16,…2, 4, 8, 16, \ldots. We can define a recursive formula for this sequence as follows:

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

To calculate the next term in the sequence, we can substitute the previous term into the formula:

f(2)=2f(1)=2(2)=4f(2) = 2f(1) = 2(2) = 4

f(3)=2f(2)=2(4)=8f(3) = 2f(2) = 2(4) = 8

f(4)=2f(3)=2(8)=16f(4) = 2f(3) = 2(8) = 16

Example 2: Recursive Formula for a Sequence with Negative Terms

Suppose we have a sequence with negative terms: −2,4,−8,16,…-2, 4, -8, 16, \ldots. We can define a recursive formula for this sequence as follows:

f(n+1)=−2f(n)f(n+1) = -2f(n)

To calculate the next term in the sequence, we can substitute the previous term into the formula:

f(2)=−2f(1)=−2(−2)=4f(2) = -2f(1) = -2(-2) = 4

f(3)=−2f(2)=−2(4)=−8f(3) = -2f(2) = -2(4) = -8

f(4)=−2f(3)=−2(−8)=16f(4) = -2f(3) = -2(-8) = 16

Conclusion

In conclusion, recursive formulas are a powerful tool for defining sequences and modeling real-world phenomena. By understanding how to determine the correct recursive formula for a given sequence, you can use this technique to solve a wide range of problems.

References

  • [1] "Recursive Sequences" by Math Open Reference
  • [2] "Recursive Formulas" by Wolfram MathWorld

Additional Resources

  • [1] "Recursive Sequences" by Khan Academy
  • [2] "Recursive Formulas" by MIT OpenCourseWare

Glossary

  • Recursive Formula: A formula that defines a sequence where each term is defined in terms of previous terms.
  • Initial Condition: The value of the first term in a sequence.
  • Sequence: A list of numbers or values that follow a specific pattern or rule.