Analyzing the Sequence of Numbers

Oct 13, 2025 by miodeya.com 46 views

Which Representation Yields the Same Outcome as the Sequence Defined Recursively?

In mathematics, sequences are an essential concept used to describe a series of numbers in a specific order. They can be defined recursively or explicitly, and understanding the relationship between these two types of definitions is crucial for solving problems in various mathematical disciplines. In this article, we will explore a sequence defined recursively and determine which representation yields the same outcome.

The given sequence is defined recursively as follows:

a_1 = 3

a_n = -4 + a_{n-1}

This means that the first term of the sequence is 3, and each subsequent term is obtained by subtracting 4 from the previous term.

We are given three explicit representations of the sequence:

Sequence of Numbers: $3, -1, -5, -9, -13, \ldots$
Linear Equation: $a_n = 4n - 1$
Another Sequence: $3, 7, 11, 15, 19, \ldots$

Analyzing the Sequence of Numbers

Let's analyze the sequence of numbers: $3, -1, -5, -9, -13, \ldots$ . We can see that each term is obtained by subtracting 4 from the previous term, which is consistent with the recursive definition.

Analyzing the Linear Equation

Now, let's analyze the linear equation: $a_n = 4n - 1$ . We can substitute the values of $n$ into this equation to obtain the corresponding terms of the sequence.

For $n = 1$ , we have:

a_1 = 4(1) - 1 = 3

For $n = 2$ , we have:

a_2 = 4(2) - 1 = -1

For $n = 3$ , we have:

a_3 = 4(3) - 1 = -5

We can continue this process to obtain the subsequent terms of the sequence.

Analyzing the Another Sequence

The another sequence is: $3, 7, 11, 15, 19, \ldots$ . We can see that each term is obtained by adding 4 to the previous term, which is not consistent with the recursive definition.

Based on our analysis, we can conclude that the sequence defined recursively is the same as the sequence of numbers: $3, -1, -5, -9, -13, \ldots$ . This is because both sequences are obtained by subtracting 4 from the previous term.

The linear equation: $a_n = 4n - 1$ also yields the same sequence, as we have shown by substituting the values of $n$ into this equation.

On the other hand, the another sequence: $3, 7, 11, 15, 19, \ldots$ is not the same as the sequence defined recursively, as it is obtained by adding 4 to the previous term.

The final answer is: $\boxed{1}$
Q&A: Understanding the Sequence Defined Recursively

In our previous article, we explored a sequence defined recursively and determined which representation yields the same outcome. In this article, we will answer some frequently asked questions related to the sequence defined recursively.

Q: What is the recursive definition of the sequence?

A: The recursive definition of the sequence is:

a_1 = 3

a_n = -4 + a_{n-1}

This means that the first term of the sequence is 3, and each subsequent term is obtained by subtracting 4 from the previous term.

Q: What is the explicit representation of the sequence?

A: The explicit representation of the sequence is:

a_n = 4n - 1

This means that each term of the sequence can be obtained by substituting the value of $n$ into this equation.

Q: How do I find the $n^{th}$ term of the sequence?

A: To find the $n^{th}$ term of the sequence, you can substitute the value of $n$ into the explicit representation of the sequence:

a_n = 4n - 1

For example, to find the 5th term of the sequence, you would substitute $n = 5$ into the equation:

a_5 = 4(5) - 1 = 19

Q: What is the pattern of the sequence?

A: The pattern of the sequence is that each term is obtained by subtracting 4 from the previous term. This is a classic example of an arithmetic sequence.

Q: Can I use the recursive definition to find the $n^{th}$ term of the sequence?

A: Yes, you can use the recursive definition to find the $n^{th}$ term of the sequence. However, this method can be time-consuming and may not be practical for large values of $n$ . The explicit representation of the sequence is generally more efficient and easier to use.

Q: What is the relationship between the recursive definition and the explicit representation of the sequence?

A: The recursive definition and the explicit representation of the sequence are equivalent. This means that they both describe the same sequence, and any term of the sequence can be obtained using either definition.

Q: Can I use the sequence defined recursively in real-world applications?

A: Yes, the sequence defined recursively has many real-world applications. For example, it can be used to model population growth, financial transactions, and other types of data that exhibit an arithmetic pattern.

In this article, we have answered some frequently asked questions related to the sequence defined recursively. We have also discussed the recursive definition, explicit representation, and pattern of the sequence, as well as its relationship to real-world applications.

The final answer is: $\boxed{1}$