Write A Recursive Formula For $a_n$, The $n^{\text{th}}$ Term Of The Sequence $7, -42, 252, -1512, \ldots$.

Introduction

In mathematics, a recursive formula is a way to define a sequence where each term is defined recursively as a function of the preceding terms. This type of formula is essential in understanding various mathematical sequences and series. In this article, we will explore how to write a recursive formula for the given sequence $7, -42, 252, -1512, \ldots$.

Understanding the Sequence

The given sequence is $7, -42, 252, -1512, \ldots$. To understand the sequence, let's examine the differences between consecutive terms:

• $-42 - 7 = -49$
• $252 - (-42) = 294$
• $-1512 - 252 = -1764$

We can observe that the differences between consecutive terms are increasing by a factor of $-6$ each time. This suggests that the sequence may be related to powers of $-6$.

Finding the Recursive Formula

To find the recursive formula, we need to identify a pattern in the sequence. Let's examine the relationship between each term and the previous term:

• $a_2 = -42 = 7 \cdot (-6)$
• $a_3 = 252 = -42 \cdot (-6)$
• $a_4 = -1512 = 252 \cdot (-6)$

We can see that each term is obtained by multiplying the previous term by $-6$. This suggests that the recursive formula for the sequence is:

$a_n = a_{n-1} \cdot (-6)$

However, this formula does not take into account the initial term $a_1 = 7$. To include the initial term, we can modify the formula as follows:

$a_n = a_{n-1} \cdot (-6)$ for $n > 1$

$a_1 = 7$

Verifying the Recursive Formula

To verify the recursive formula, let's calculate the first few terms of the sequence using the formula:

• $a_1 = 7$
• $a_2 = a_1 \cdot (-6) = 7 \cdot (-6) = -42$
• $a_3 = a_2 \cdot (-6) = -42 \cdot (-6) = 252$
• $a_4 = a_3 \cdot (-6) = 252 \cdot (-6) = -1512$

We can see that the calculated terms match the given sequence. This verifies that the recursive formula is correct.

Conclusion

In this article, we have explored how to write a recursive formula for the given sequence $7, -42, 252, -1512, \ldots$. We have identified a pattern in the sequence and used it to derive the recursive formula. The recursive formula is:

$a_n = a_{n-1} \cdot (-6)$ for $n > 1$

$a_1 = 7$

This formula allows us to calculate any term of the sequence using the previous term. We have also verified the recursive formula by calculating the first few terms of the sequence.

Applications of Recursive Formulas

Recursive formulas have numerous applications in mathematics and other fields. Some examples include:

• Mathematical sequences and series: Recursive formulas are used to define and analyze various mathematical sequences and series, such as the Fibonacci sequence and the harmonic series.
• Computer science: Recursive formulas are used in computer science to solve problems such as tree traversals and graph algorithms.
• Economics: Recursive formulas are used in economics to model economic systems and make predictions about future economic trends.

Common Mistakes to Avoid

When working with recursive formulas, it's essential to avoid common mistakes such as:

• Incorrect initial conditions: Make sure to specify the initial conditions correctly, as they can affect the entire sequence.
• Incorrect recursive formula: Double-check the recursive formula to ensure it's correct and consistent with the sequence.
• Infinite recursion: Be careful not to create an infinite recursion, which can lead to a stack overflow error.

Conclusion

Introduction

In our previous article, we explored how to write a recursive formula for the given sequence $7, -42, 252, -1512, \ldots$. We identified a pattern in the sequence and used it to derive the recursive formula. In this article, we will answer some frequently asked questions about recursive formulas and provide additional insights into their applications.

Q&A

Q: What is a recursive formula?

A: A recursive formula is a way to define a sequence where each term is defined recursively as a function of the preceding terms. It's a formula that uses previous terms to calculate the next term in the sequence.

Q: How do I know if a sequence has a recursive formula?

A: To determine if a sequence has a recursive formula, look for a pattern in the differences between consecutive terms. If the differences are increasing or decreasing by a consistent factor, it may indicate a recursive formula.

Q: What are some common types of recursive formulas?

A: There are several types of recursive formulas, including:

• Linear recursive formulas: These formulas involve a linear relationship between consecutive terms, such as $a_n = a_{n-1} + c$.
• Quadratic recursive formulas: These formulas involve a quadratic relationship between consecutive terms, such as $a_n = a_{n-1}^2 + c$.
• Exponential recursive formulas: These formulas involve an exponential relationship between consecutive terms, such as $a_n = a_{n-1} \cdot c$.

Q: How do I verify a recursive formula?

A: To verify a recursive formula, calculate the first few terms of the sequence using the formula and compare them to the given sequence. If the calculated terms match the given sequence, the recursive formula is correct.

Q: What are some applications of recursive formulas?

A: Recursive formulas have numerous applications in mathematics and other fields, including:

• Mathematical sequences and series: Recursive formulas are used to define and analyze various mathematical sequences and series, such as the Fibonacci sequence and the harmonic series.
• Computer science: Recursive formulas are used in computer science to solve problems such as tree traversals and graph algorithms.
• Economics: Recursive formulas are used in economics to model economic systems and make predictions about future economic trends.

Q: What are some common mistakes to avoid when working with recursive formulas?

A: When working with recursive formulas, it's essential to avoid common mistakes such as:

• Incorrect initial conditions: Make sure to specify the initial conditions correctly, as they can affect the entire sequence.
• Incorrect recursive formula: Double-check the recursive formula to ensure it's correct and consistent with the sequence.
• Infinite recursion: Be careful not to create an infinite recursion, which can lead to a stack overflow error.

Q: How do I write a recursive formula for a given sequence?

A: To write a recursive formula for a given sequence, follow these steps:

1. Examine the sequence: Look for a pattern in the differences between consecutive terms.
2. Identify the relationship: Determine the relationship between consecutive terms, such as a linear or exponential relationship.
3. Write the recursive formula: Use the identified relationship to write a recursive formula for the sequence.

Conclusion

In conclusion, recursive formulas are a powerful tool for defining and analyzing mathematical sequences and series. By understanding how to write a recursive formula and avoiding common mistakes, we can gain insights into the underlying structure of the sequence and make predictions about future terms. With practice and experience, we can develop the skills to work with recursive formulas and apply them to various problems in mathematics and other fields.

Additional Resources

For further learning, we recommend the following resources:

• Mathematical textbooks: Consult mathematical textbooks, such as "Introduction to Algorithms" by Thomas H. Cormen, for a comprehensive introduction to recursive formulas.
• Online resources: Visit online resources, such as Khan Academy and MIT OpenCourseWare, for video lectures and tutorials on recursive formulas.
• Practice problems: Practice solving problems involving recursive formulas to develop your skills and confidence.