Three Terms Of An Arithmetic Sequence Are Shown Below. Which Recursive Formula Defines The Sequence?1. $f(1)=6$, $f(4)=12$, $f(7)=18$A. $f(n+1)=f(n)+6$ B. $f(n+1)=2f(n$\] C. $f(n+1)=f(n)+2$ D.

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Introduction

Arithmetic sequences are a fundamental concept in mathematics, and understanding their properties is crucial for solving various problems in mathematics and other fields. A recursive formula is a way to define a sequence by describing how each term is related to the previous term. In this article, we will explore how to identify the recursive formula for an arithmetic sequence given three terms.

What is an Arithmetic Sequence?

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference. For example, the sequence 2, 5, 8, 11, ... is an arithmetic sequence with a common difference of 3.

Recursive Formulas

A recursive formula is a way to define a sequence by describing how each term is related to the previous term. For an arithmetic sequence, the recursive formula can be written in the form:

f(n+1) = f(n) + d

where f(n) is the nth term of the sequence, and d is the common difference.

Given Terms

We are given three terms of an arithmetic sequence: f(1) = 6, f(4) = 12, and f(7) = 18. We need to find the recursive formula that defines this sequence.

Analyzing the Given Terms

Let's analyze the given terms to find the common difference. We can see that the difference between f(4) and f(1) is 12 - 6 = 6, and the difference between f(7) and f(4) is 18 - 12 = 6. This suggests that the common difference is 6.

Finding the Recursive Formula

Now that we have found the common difference, we can write the recursive formula for the sequence. Since the common difference is 6, we can write:

f(n+1) = f(n) + 6

This is the recursive formula that defines the sequence.

Conclusion

In this article, we have explored how to identify the recursive formula for an arithmetic sequence given three terms. We analyzed the given terms to find the common difference and then wrote the recursive formula for the sequence. The recursive formula is a powerful tool for defining sequences and is widely used in mathematics and other fields.

Final Answer

The final answer is:

A. f(n+1) = f(n) + 6

Discussion

This problem is a great example of how to apply mathematical concepts to real-world problems. The recursive formula is a fundamental concept in mathematics, and understanding it is crucial for solving various problems in mathematics and other fields. The problem also requires critical thinking and analysis to find the common difference and write the recursive formula.

Related Topics

  • Arithmetic sequences
  • Recursive formulas
  • Common difference
  • Sequence definition

References

  • [1] "Arithmetic Sequences and Series" by Math Open Reference
  • [2] "Recursive Formulas" by Math Is Fun
  • [3] "Arithmetic Sequences" by Khan Academy
    Q&A: Understanding Recursive Formulas in Arithmetic Sequences ===========================================================

Introduction

In our previous article, we explored how to identify the recursive formula for an arithmetic sequence given three terms. In this article, we will answer some frequently asked questions about recursive formulas in arithmetic sequences.

Q: What is a recursive formula?

A: A recursive formula is a way to define a sequence by describing how each term is related to the previous term. For an arithmetic sequence, the recursive formula can be written in the form:

f(n+1) = f(n) + d

where f(n) is the nth term of the sequence, and d is the common difference.

Q: How do I find the recursive formula for an arithmetic sequence?

A: To find the recursive formula for an arithmetic sequence, you need to find the common difference between the terms. You can do this by subtracting each term from the previous term. Once you have found the common difference, you can write the recursive formula in the form:

f(n+1) = f(n) + d

Q: What is the common difference in an arithmetic sequence?

A: The common difference is the constant difference between any two consecutive terms in an arithmetic sequence. For example, in the sequence 2, 5, 8, 11, ..., the common difference is 3.

Q: How do I determine if a sequence is arithmetic or not?

A: To determine if a sequence is arithmetic or not, you need to check if the difference between any two consecutive terms is constant. If the difference is constant, then the sequence is arithmetic.

Q: Can a recursive formula have a negative common difference?

A: Yes, a recursive formula can have a negative common difference. For example, the sequence -2, -5, -8, -11, ... has a common difference of -3.

Q: Can a recursive formula have a zero common difference?

A: Yes, a recursive formula can have a zero common difference. For example, the sequence 2, 2, 2, 2, ... has a common difference of 0.

Q: How do I use a recursive formula to find the nth term of an arithmetic sequence?

A: To use a recursive formula to find the nth term of an arithmetic sequence, you need to start with the first term and apply the recursive formula repeatedly until you reach the nth term. For example, if the recursive formula is:

f(n+1) = f(n) + 3

and the first term is f(1) = 2, then the second term is f(2) = f(1) + 3 = 5, the third term is f(3) = f(2) + 3 = 8, and so on.

Q: Can a recursive formula be used to define a geometric sequence?

A: No, a recursive formula cannot be used to define a geometric sequence. A recursive formula is used to define an arithmetic sequence, while a geometric sequence is defined by a formula that involves multiplication.

Conclusion

In this article, we have answered some frequently asked questions about recursive formulas in arithmetic sequences. We hope that this article has provided you with a better understanding of recursive formulas and how to use them to define arithmetic sequences.

Final Answer

The final answer is:

  • A recursive formula is a way to define a sequence by describing how each term is related to the previous term.
  • The common difference is the constant difference between any two consecutive terms in an arithmetic sequence.
  • A recursive formula can have a negative or zero common difference.
  • A recursive formula can be used to find the nth term of an arithmetic sequence.
  • A recursive formula cannot be used to define a geometric sequence.

Discussion

This article is a great resource for anyone who wants to learn more about recursive formulas and arithmetic sequences. The questions and answers provide a clear and concise explanation of the concepts, and the examples help to illustrate the ideas.

Related Topics

  • Arithmetic sequences
  • Recursive formulas
  • Common difference
  • Sequence definition

References

  • [1] "Arithmetic Sequences and Series" by Math Open Reference
  • [2] "Recursive Formulas" by Math Is Fun
  • [3] "Arithmetic Sequences" by Khan Academy