The Third Term In An Arithmetic Sequence Is 3, And The Common Difference Between Each Term Is 4. Write A Recursive Formula For The Sequence:A. A N = 4 A N − 1 A_n = 4a_{n-1} A N ​ = 4 A N − 1 ​ Where A 3 = 3 A_3 = 3 A 3 ​ = 3 B. A N = A N − 1 + 4 A_n = A_{n-1} + 4 A N ​ = A N − 1 ​ + 4 Where A 3 = 3 A_3 = 3 A 3 ​ = 3

Introduction

Arithmetic sequences are a fundamental concept in mathematics, and understanding their properties is crucial for solving various problems in algebra, geometry, and other branches of mathematics. In this article, we will explore the concept of arithmetic sequences, particularly focusing on the third term in an arithmetic sequence with a common difference of 4. We will derive a recursive formula for the sequence and discuss its implications.

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, 6, 10, 14, ... is an arithmetic sequence with a common difference of 4.

The Third Term in an Arithmetic Sequence

Given that the third term in an arithmetic sequence is 3, and the common difference between each term is 4, we can use this information to derive a recursive formula for the sequence.

Recursive Formula for the Sequence

A recursive formula for a sequence is a formula that defines each term in the sequence as a function of the previous term(s). In the case of an arithmetic sequence, the recursive formula can be written as:

$a_n = a_{n-1} + d$

where $a_n$ is the nth term in the sequence, $a_{n-1}$ is the (n-1)th term, and $d$ is the common difference.

Deriving the Recursive Formula

To derive the recursive formula for the given sequence, we can start by writing the first few terms of the sequence:

$a_1$, $a_2$, $a_3$, ...

Since the third term is 3, we can write:

$a_3 = 3$

Using the recursive formula, we can write:

$a_3 = a_2 + d$

Substituting $a_3 = 3$ and $d = 4$, we get:

$3 = a_2 + 4$

Solving for $a_2$, we get:

$a_2 = -1$

Now, we can use the recursive formula again to find $a_1$:

$a_2 = a_1 + d$

Substituting $a_2 = -1$ and $d = 4$, we get:

$-1 = a_1 + 4$

Solving for $a_1$, we get:

$a_1 = -5$

Conclusion

In this article, we derived a recursive formula for an arithmetic sequence with a third term of 3 and a common difference of 4. The recursive formula is:

$a_n = a_{n-1} + 4$

where $a_n$ is the nth term in the sequence, and $a_{n-1}$ is the (n-1)th term.

Discussion

The recursive formula for an arithmetic sequence can be used to find any term in the sequence, given the value of the previous term. This is particularly useful when working with large sequences or when the sequence is defined recursively.

Example Use Case

Suppose we want to find the 10th term in the sequence. We can use the recursive formula to find the 10th term, given the value of the 9th term.

$a_{10} = a_9 + 4$

To find $a_9$, we can use the recursive formula again:

$a_9 = a_8 + 4$

Continuing this process, we can find the value of $a_9$ and then use it to find $a_{10}$.

Conclusion

In conclusion, the recursive formula for an arithmetic sequence is a powerful tool for finding any term in the sequence, given the value of the previous term. By understanding the properties of arithmetic sequences and how to derive recursive formulas, we can solve a wide range of problems in mathematics and other fields.

References

• [1] "Arithmetic Sequences" by Math Open Reference
• [2] "Recursive Formulas" by Khan Academy

Further Reading

• "Arithmetic Sequences and Series" by Paul's Online Math Notes
• "Recursive Sequences" by Wolfram MathWorld
  The Third Term in an Arithmetic Sequence: Q&A =====================================================

Introduction

In our previous article, we explored the concept of arithmetic sequences, particularly focusing on the third term in an arithmetic sequence with a common difference of 4. We derived a recursive formula for the sequence and discussed its implications. In this article, we will answer some frequently asked questions (FAQs) related to arithmetic sequences and the third term in an arithmetic sequence.

Q&A

Q: What is an arithmetic sequence?

A: 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.

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.

Q: How do I find the nth term in an arithmetic sequence?

A: To find the nth term in an arithmetic sequence, you can use the recursive formula:

$a_n = a_{n-1} + d$

where $a_n$ is the nth term in the sequence, $a_{n-1}$ is the (n-1)th term, and $d$ is the common difference.

Q: What is the recursive formula for an arithmetic sequence?

A: The recursive formula for an arithmetic sequence is:

$a_n = a_{n-1} + d$

where $a_n$ is the nth term in the sequence, $a_{n-1}$ is the (n-1)th term, and $d$ is the common difference.

Q: How do I find the first term in an arithmetic sequence?

A: To find the first term in an arithmetic sequence, you can use the recursive formula:

$a_1 = a_2 - d$

where $a_1$ is the first term in the sequence, $a_2$ is the second term, and $d$ is the common difference.

Q: What is the formula for the sum of an arithmetic sequence?

A: The formula for the sum of an arithmetic sequence is:

$S_n = \frac{n}{2}(a_1 + a_n)$

where $S_n$ is the sum of the first n terms, $a_1$ is the first term, $a_n$ is the nth term, and $n$ is the number of terms.

Q: How do I find the sum of an arithmetic sequence?

A: To find the sum of an arithmetic sequence, you can use the formula:

$S_n = \frac{n}{2}(a_1 + a_n)$

where $S_n$ is the sum of the first n terms, $a_1$ is the first term, $a_n$ is the nth term, and $n$ is the number of terms.

Q: What is the formula for the nth term of an arithmetic sequence?

A: The formula for the nth term of an arithmetic sequence is:

$a_n = a_1 + (n-1)d$

where $a_n$ is the nth term, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference.

Q: How do I find the nth term of an arithmetic sequence?

A: To find the nth term of an arithmetic sequence, you can use the formula:

$a_n = a_1 + (n-1)d$

where $a_n$ is the nth term, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference.

Conclusion

In this article, we answered some frequently asked questions (FAQs) related to arithmetic sequences and the third term in an arithmetic sequence. We hope that this article has provided you with a better understanding of arithmetic sequences and how to work with them.

References

• [1] "Arithmetic Sequences" by Math Open Reference
• [2] "Recursive Formulas" by Khan Academy
• [3] "Arithmetic Sequences and Series" by Paul's Online Math Notes
• [4] "Recursive Sequences" by Wolfram MathWorld

Further Reading

• "Arithmetic Sequences and Series" by Paul's Online Math Notes
• "Recursive Sequences" by Wolfram MathWorld
• "Arithmetic Sequences" by Math Is Fun