Determine The Sequence Defined By The Following Formulas:${ \begin{array}{l} a_1 = 3 \ a_{n+1} = -2 \cdot A_n \end{array} }$

Introduction

In mathematics, a recurrence relation is a way to define a sequence where each term is defined recursively as a function of previous terms. In this article, we will explore a specific recurrence relation and determine the sequence it defines. The recurrence relation is given by:

$$a_1 = 3$$

$$a_{n+1} = -2 \cdot a_n$$

Understanding the Recurrence Relation

The recurrence relation is defined for a sequence $a_n$, where $n$ is a positive integer. The first term of the sequence, $a_1$, is given as 3. The subsequent terms of the sequence are defined recursively as $a_{n+1} = -2 \cdot a_n$. This means that each term is obtained by multiplying the previous term by -2.

Calculating the First Few Terms

To understand the sequence, let's calculate the first few terms:

• $a_1 = 3$
• $a_2 = -2 \cdot a_1 = -2 \cdot 3 = -6$
• $a_3 = -2 \cdot a_2 = -2 \cdot (-6) = 12$
• $a_4 = -2 \cdot a_3 = -2 \cdot 12 = -24$
• $a_5 = -2 \cdot a_4 = -2 \cdot (-24) = 48$

Observations

From the first few terms, we can observe that the sequence is alternating between positive and negative terms. The absolute value of each term is increasing by a factor of 2.

Generalizing the Sequence

To generalize the sequence, let's analyze the pattern:

• The first term is 3.
• The second term is -2 times the first term.
• The third term is -2 times the second term, which is 2 times the first term.
• The fourth term is -2 times the third term, which is 4 times the first term.
• The fifth term is -2 times the fourth term, which is 8 times the first term.

The General Formula

Based on the pattern, we can conclude that the nth term of the sequence is given by:

$$a_n = (-2)^{n-1} \cdot 3$$

Proof of the General Formula

To prove the general formula, we can use mathematical induction.

Base Case

The base case is when n = 1. In this case, the formula gives:

$$a_1 = (-2)^{1-1} \cdot 3 = 3$$

This matches the given initial condition.

Inductive Step

Assume that the formula holds for some positive integer k. That is:

$$a_k = (-2)^{k-1} \cdot 3$$

We need to show that the formula holds for k + 1. That is:

$$a_{k+1} = (-2)^{k} \cdot 3$$

Using the recurrence relation, we have:

$$a_{k+1} = -2 \cdot a_k = -2 \cdot (-2)^{k-1} \cdot 3 = (-2)^k \cdot 3$$

This shows that the formula holds for k + 1.

Conclusion

We have determined the sequence defined by the recurrence relation. The general formula for the nth term is:

$$a_n = (-2)^{n-1} \cdot 3$$

This formula can be used to calculate any term of the sequence.

Applications

The sequence defined by the recurrence relation has several applications in mathematics and computer science. For example, it can be used to model population growth or decay, or to analyze the behavior of recursive algorithms.

Further Reading

For further reading on recurrence relations and their applications, we recommend the following resources:

• "Introduction to Recurrence Relations" by MIT OpenCourseWare
• "Recurrence Relations" by Wolfram MathWorld
• "Recurrence Relations" by Khan Academy

References

• [1] "Recurrence Relations" by David M. Bressoud, in "A Course in Computational Algebraic Number Theory" (Springer, 1991)
• [2] "Recurrence Relations" by Ronald L. Graham, in "Concrete Mathematics" (Addison-Wesley, 1989)

Glossary

• Recurrence relation: A way to define a sequence where each term is defined recursively as a function of previous terms.
• Sequence: A list of numbers or objects that are related to each other in a specific way.
• Term: A single element of a sequence.
• Recursive formula: A formula that defines a sequence recursively, using previous terms to define the next term.
  Frequently Asked Questions (FAQs) about Recurrence Relations ================================================================

Q: What is a recurrence relation?

A: A recurrence relation is a way to define a sequence where each term is defined recursively as a function of previous terms. It is a mathematical formula that defines a sequence recursively, using previous terms to define the next term.

Q: What is the difference between a recurrence relation and a recursive formula?

A: A recurrence relation is a general term that refers to any mathematical formula that defines a sequence recursively. A recursive formula is a specific type of recurrence relation that defines a sequence using a fixed number of previous terms.

Q: How do I determine the sequence defined by a recurrence relation?

A: To determine the sequence defined by a recurrence relation, you need to:

1. Identify the initial term(s) of the sequence.
2. Identify the recurrence relation that defines the sequence.
3. Use the recurrence relation to calculate the next term in the sequence.
4. Repeat step 3 until you have calculated the desired number of terms.

Q: What is the general formula for a sequence defined by a recurrence relation?

A: The general formula for a sequence defined by a recurrence relation is:

$$a_n = f(a_{n-1}, a_{n-2}, ..., a_1)$$

where $f$ is the recurrence relation and $a_n$ is the nth term of the sequence.

Q: How do I prove the general formula for a sequence defined by a recurrence relation?

A: To prove the general formula for a sequence defined by a recurrence relation, you need to use mathematical induction. This involves:

1. Proving the base case, which is the initial term(s) of the sequence.
2. Assuming that the formula holds for some positive integer k.
3. Proving that the formula holds for k + 1.

Q: What are some common types of recurrence relations?

A: Some common types of recurrence relations include:

• Linear recurrence relations: These are recurrence relations of the form $a_n = c_1 a_{n-1} + c_2 a_{n-2} + ... + c_k a_{n-k}$.
• Nonlinear recurrence relations: These are recurrence relations that do not fit the linear form.
• Homogeneous recurrence relations: These are recurrence relations of the form $a_n = c_1 a_{n-1} + c_2 a_{n-2} + ... + c_k a_{n-k}$, where $c_1, c_2, ..., c_k$ are constants.
• Nonhomogeneous recurrence relations: These are recurrence relations that do not fit the homogeneous form.

Q: What are some applications of recurrence relations?

A: Recurrence relations have many applications in mathematics and computer science, including:

• Modeling population growth or decay.
• Analyzing the behavior of recursive algorithms.
• Solving problems in combinatorics and graph theory.
• Modeling financial systems and economic models.

Q: How do I use recurrence relations in real-world problems?

A: To use recurrence relations in real-world problems, you need to:

1. Identify the problem and the variables involved.
2. Develop a mathematical model of the problem using recurrence relations.
3. Solve the recurrence relation to obtain the desired solution.
4. Interpret the results in the context of the problem.

Q: What are some common mistakes to avoid when working with recurrence relations?

A: Some common mistakes to avoid when working with recurrence relations include:

• Failing to identify the initial term(s) of the sequence.
• Failing to identify the recurrence relation that defines the sequence.
• Making errors in the calculation of the next term in the sequence.
• Failing to prove the general formula for the sequence.

Q: How do I learn more about recurrence relations?

A: To learn more about recurrence relations, you can:

• Read books and articles on the subject.
• Take online courses or attend lectures on recurrence relations.
• Practice solving problems and exercises on recurrence relations.
• Join online communities or forums to discuss recurrence relations with others.