If $a_1 = 4$ And $a_n = 4a_{n-1}$, Then Find The Value Of $ A 4 A_4 A 4 ​ [/tex].

Introduction to Recursive Sequences

Recursive sequences are a fundamental concept in mathematics, where each term is defined recursively as a function of the preceding term. In this article, we will explore a specific recursive sequence and find the value of its fourth term.

The Recursive Sequence

The given recursive sequence is defined as follows:

$$a_1 = 4$$

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

This means that each term is four times the previous term. To find the value of $a_4$, we need to find the values of $a_2$ and $a_3$ first.

Finding the Value of $a_2$

To find the value of $a_2$, we substitute $n = 2$ into the recursive formula:

$$a_2 = 4a_{2-1}$$

$$a_2 = 4a_1$$

Since $a_1 = 4$, we have:

$$a_2 = 4(4)$$

$$a_2 = 16$$

Finding the Value of $a_3$

To find the value of $a_3$, we substitute $n = 3$ into the recursive formula:

$$a_3 = 4a_{3-1}$$

$$a_3 = 4a_2$$

Since $a_2 = 16$, we have:

$$a_3 = 4(16)$$

$$a_3 = 64$$

Finding the Value of $a_4$

To find the value of $a_4$, we substitute $n = 4$ into the recursive formula:

$$a_4 = 4a_{4-1}$$

$$a_4 = 4a_3$$

Since $a_3 = 64$, we have:

$$a_4 = 4(64)$$

$$a_4 = 256$$

Conclusion

In this article, we explored a recursive sequence defined by the formula $a_n = 4a_{n-1}$, with the initial term $a_1 = 4$. We found the values of $a_2$, $a_3$, and $a_4$ by substituting the values of the previous terms into the recursive formula. The value of $a_4$ is 256.

Understanding the Pattern

The recursive sequence $a_n = 4a_{n-1}$ is a geometric sequence, where each term is obtained by multiplying the previous term by a fixed constant, in this case, 4. The pattern of the sequence is:

$$a_1 = 4$$

$$a_2 = 4(4) = 16$$

$$a_3 = 4(16) = 64$$

$$a_4 = 4(64) = 256$$

Generalizing the Recursive Sequence

The recursive sequence $a_n = 4a_{n-1}$ can be generalized to any initial term $a_1$ and any common ratio $r$. The general formula for the $n$th term is:

$$a_n = ra_{n-1}$$

Applications of Recursive Sequences

Recursive sequences have numerous applications in mathematics, computer science, and engineering. Some examples include:

• Fibonacci sequence: A recursive sequence where each term is the sum of the two preceding terms.
• Binomial coefficients: A recursive sequence used to calculate the number of combinations of a set of objects.
• Dynamic programming: A method for solving complex problems by breaking them down into smaller subproblems.

Conclusion

In this article, we explored a recursive sequence defined by the formula $a_n = 4a_{n-1}$, with the initial term $a_1 = 4$. We found the values of $a_2$, $a_3$, and $a_4$ by substituting the values of the previous terms into the recursive formula. The value of $a_4$ is 256. We also discussed the pattern of the sequence, generalized the recursive sequence, and provided examples of applications of recursive sequences.

Introduction

In our previous article, we explored a recursive sequence defined by the formula $a_n = 4a_{n-1}$, with the initial term $a_1 = 4$. We found the values of $a_2$, $a_3$, and $a_4$ by substituting the values of the previous terms into the recursive formula. In this article, we will answer some frequently asked questions about recursive sequences.

Q: What is a recursive sequence?

A: A recursive sequence is a sequence of numbers where each term is defined recursively as a function of the preceding term. In other words, each term is obtained by applying a recursive formula to the previous term.

Q: What is the difference between a recursive sequence and a recursive function?

A: A recursive sequence is a sequence of numbers, while a recursive function is a function that calls itself to compute its value. While both concepts involve recursion, they are distinct and serve different purposes.

Q: How do I find the value of a term in a recursive sequence?

A: To find the value of a term in a recursive sequence, you need to substitute the value of the previous term into the recursive formula. For example, if the recursive formula is $a_n = 4a_{n-1}$, you would substitute the value of $a_{n-1}$ into the formula to get the value of $a_n$.

Q: What is the pattern of a recursive sequence?

A: The pattern of a recursive sequence depends on the recursive formula. In the case of the sequence $a_n = 4a_{n-1}$, the pattern is a geometric sequence, where each term is obtained by multiplying the previous term by a fixed constant, in this case, 4.

Q: Can I use a recursive sequence to model real-world phenomena?

A: Yes, recursive sequences can be used to model real-world phenomena, such as population growth, financial transactions, and more. The key is to identify the recursive pattern in the data and use it to make predictions or model the behavior of the system.

Q: How do I determine the initial term of a recursive sequence?

A: The initial term of a recursive sequence is typically given or can be determined from the problem statement. In some cases, you may need to use additional information or constraints to determine the initial term.

Q: Can I use a recursive sequence to solve a problem that has multiple solutions?

A: Yes, recursive sequences can be used to solve problems with multiple solutions. The key is to identify the recursive pattern in the data and use it to find all possible solutions.

Q: What are some common applications of recursive sequences?

A: Recursive sequences have numerous applications in mathematics, computer science, and engineering, including:

• Fibonacci sequence: A recursive sequence where each term is the sum of the two preceding terms.
• Binomial coefficients: A recursive sequence used to calculate the number of combinations of a set of objects.
• Dynamic programming: A method for solving complex problems by breaking them down into smaller subproblems.
• Population growth: A recursive sequence used to model population growth and make predictions about future population sizes.

Conclusion

In this article, we answered some frequently asked questions about recursive sequences. We discussed the definition of a recursive sequence, how to find the value of a term, the pattern of a recursive sequence, and some common applications of recursive sequences. We hope this article has provided you with a better understanding of recursive sequences and how they can be used to solve problems in mathematics, computer science, and engineering.