Kina Spends 1.5 Hours Setting Up Her Sewing Machine And Making One Hat. The Total Amount Of Time Spent Making Hats Can Be Represented By The Sequence Below: 1.5 , 2.25 , 3.0 , 3.75 , … 1.5, 2.25, 3.0, 3.75, \ldots 1.5 , 2.25 , 3.0 , 3.75 , … Which Recursive Formula Can Be Used To Determine The

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Introduction

Kina is a skilled sewer who spends a significant amount of time setting up her machine and creating hats. The time spent making hats follows a specific sequence: 1.5, 2.25, 3.0, 3.75, and so on. In this article, we will explore the recursive formula that can be used to determine the total amount of time spent making hats.

The Sequence: A Closer Look

The given sequence represents the time spent making hats by Kina. To understand the pattern, let's examine the differences between consecutive terms:

  • 2.25 - 1.5 = 0.75
  • 3.0 - 2.25 = 0.75
  • 3.75 - 3.0 = 0.75

As we can see, the differences between consecutive terms are constant, which indicates that the sequence is arithmetic. However, the first term is not the typical starting point for an arithmetic sequence. Let's re-examine the sequence and try to find a more suitable starting point.

Re-examining the Sequence

If we start from the second term (2.25), we can see that the sequence becomes:

  • 2.25, 3.0, 3.75, ...

Now, let's calculate the differences between consecutive terms:

  • 3.0 - 2.25 = 0.75
  • 3.75 - 3.0 = 0.75

The differences are still constant, which confirms that the sequence is arithmetic. However, we still need to find the recursive formula that can be used to determine the total amount of time spent making hats.

The Recursive Formula

Let's denote the time spent making hats as t_n, where n represents the number of hats made. We can see that each term is obtained by adding 0.75 to the previous term. Therefore, the recursive formula can be written as:

t_n = t_(n-1) + 0.75

This formula states that the time spent making the n-th hat is equal to the time spent making the (n-1)-th hat plus 0.75.

Example

Let's use the recursive formula to calculate the time spent making the first few hats:

  • t_1 = 1.5 (given)
  • t_2 = t_1 + 0.75 = 1.5 + 0.75 = 2.25
  • t_3 = t_2 + 0.75 = 2.25 + 0.75 = 3.0
  • t_4 = t_3 + 0.75 = 3.0 + 0.75 = 3.75

As we can see, the recursive formula produces the correct values for the sequence.

Conclusion

In this article, we have explored the recursive formula that can be used to determine the total amount of time spent making hats by Kina. The formula is based on the arithmetic sequence, where each term is obtained by adding 0.75 to the previous term. We have also provided an example to demonstrate the use of the recursive formula. By understanding the recursive formula, we can calculate the time spent making hats for any given number of hats.

References

Glossary

  • Arithmetic Sequence: A sequence of numbers in which the difference between consecutive terms is constant.
  • Recursive Formula: A formula that defines a sequence recursively, where each term is defined in terms of the previous term(s).
    Kina's Hats: A Recursive Formula Q&A =====================================

Introduction

In our previous article, we explored the recursive formula that can be used to determine the total amount of time spent making hats by Kina. In this article, we will answer some frequently asked questions about the recursive formula and provide additional insights into the problem.

Q: What is the recursive formula for Kina's hats?

A: The recursive formula for Kina's hats is:

t_n = t_(n-1) + 0.75

This formula states that the time spent making the n-th hat is equal to the time spent making the (n-1)-th hat plus 0.75.

Q: How does the recursive formula work?

A: The recursive formula works by starting with the initial value of t_1 = 1.5 and then repeatedly applying the formula to calculate the next term in the sequence. For example, to calculate t_2, we add 0.75 to t_1, resulting in t_2 = 2.25. To calculate t_3, we add 0.75 to t_2, resulting in t_3 = 3.0, and so on.

Q: What is the pattern in the sequence?

A: The pattern in the sequence is an arithmetic sequence, where each term is obtained by adding 0.75 to the previous term. This means that the difference between consecutive terms is constant, which is a characteristic of arithmetic sequences.

Q: Can I use the recursive formula to calculate the time spent making any number of hats?

A: Yes, you can use the recursive formula to calculate the time spent making any number of hats. Simply start with the initial value of t_1 = 1.5 and then repeatedly apply the formula to calculate the next term in the sequence.

Q: How do I know when to stop applying the recursive formula?

A: You can stop applying the recursive formula when you have calculated the time spent making the desired number of hats. For example, if you want to calculate the time spent making 10 hats, you would apply the formula 10 times, starting with t_1 = 1.5 and ending with t_10.

Q: Can I use the recursive formula to calculate the time spent making hats for a different initial value?

A: Yes, you can use the recursive formula to calculate the time spent making hats for a different initial value. Simply replace the initial value of t_1 = 1.5 with the desired initial value and then apply the formula as usual.

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

A: The recursive formula has many real-world applications, including:

  • Calculating the time spent on a project with a fixed sequence of tasks
  • Modeling population growth or decline in a population with a fixed birth and death rate
  • Calculating the cost of a product with a fixed sequence of components

Conclusion

In this article, we have answered some frequently asked questions about the recursive formula for Kina's hats and provided additional insights into the problem. We hope that this article has been helpful in understanding the recursive formula and its applications.

References

Glossary

  • Arithmetic Sequence: A sequence of numbers in which the difference between consecutive terms is constant.
  • Recursive Formula: A formula that defines a sequence recursively, where each term is defined in terms of the previous term(s).
  • Initial Value: The first term in a sequence, which is used as the starting point for the recursive formula.