Which Answer Is The Explicit Rule For The Sequence $5, 11, 17, 23, 29, \ldots$?A. $a_n = 1 - 6n$B. $a_n = 1 + 6n$C. $a_n = -1 + 6n$D. $a_n = -1 - 6n$

Introduction

In mathematics, a sequence is a list of numbers in a specific order. Identifying the explicit rule for a sequence is crucial in understanding its behavior and making predictions about future terms. The given sequence is $5, 11, 17, 23, 29, \ldots$, and we need to determine which of the provided options is the explicit rule for this sequence.

Understanding the Sequence

The sequence provided is an arithmetic sequence, where each term is obtained by adding a fixed constant to the previous term. To identify the explicit rule, we need to find the common difference between consecutive terms. Let's calculate the differences between consecutive terms:

• $11 - 5 = 6$
• $17 - 11 = 6$
• $23 - 17 = 6$
• $29 - 23 = 6$

As we can see, the common difference between consecutive terms is $6$. This means that each term in the sequence is obtained by adding $6$ to the previous term.

Analyzing the Options

Now that we have identified the common difference, let's analyze the provided options:

A. $a_n = 1 - 6n$ B. $a_n = 1 + 6n$ C. $a_n = -1 + 6n$ D. $a_n = -1 - 6n$

We need to determine which of these options matches the sequence $5, 11, 17, 23, 29, \ldots$.

Option A: $a_n = 1 - 6n$

Let's substitute the first term $a_1 = 5$ into the equation $a_n = 1 - 6n$:

$$5 = 1 - 6(1)$$

Simplifying the equation, we get:

$$5 = 1 - 6$$

This is not true, as $5 \neq -5$. Therefore, option A is not the explicit rule for the sequence.

Option B: $a_n = 1 + 6n$

Let's substitute the first term $a_1 = 5$ into the equation $a_n = 1 + 6n$:

$$5 = 1 + 6(1)$$

Simplifying the equation, we get:

$$5 = 1 + 6$$

This is true, as $5 = 7 - 2$. Therefore, option B is a potential candidate for the explicit rule.

Option C: $a_n = -1 + 6n$

Let's substitute the first term $a_1 = 5$ into the equation $a_n = -1 + 6n$:

$$5 = -1 + 6(1)$$

Simplifying the equation, we get:

$$5 = -1 + 6$$

This is true, as $5 = 5$. Therefore, option C is also a potential candidate for the explicit rule.

Option D: $a_n = -1 - 6n$

Let's substitute the first term $a_1 = 5$ into the equation $a_n = -1 - 6n$:

$$5 = -1 - 6(1)$$

Simplifying the equation, we get:

$$5 = -1 - 6$$

This is not true, as $5 \neq -7$. Therefore, option D is not the explicit rule for the sequence.

Conclusion

Based on our analysis, we have two potential candidates for the explicit rule: options B and C. Let's further analyze these options to determine which one is the correct explicit rule.

Option B: $a_n = 1 + 6n$

Let's calculate the first few terms of the sequence using the equation $a_n = 1 + 6n$:

• $a_1 = 1 + 6(1) = 7$
• $a_2 = 1 + 6(2) = 13$
• $a_3 = 1 + 6(3) = 19$
• $a_4 = 1 + 6(4) = 25$
• $a_5 = 1 + 6(5) = 31$

As we can see, the sequence generated by option B is $7, 13, 19, 25, 31, \ldots$, which is not the same as the original sequence $5, 11, 17, 23, 29, \ldots$.

Option C: $a_n = -1 + 6n$

Let's calculate the first few terms of the sequence using the equation $a_n = -1 + 6n$:

• $a_1 = -1 + 6(1) = 5$
• $a_2 = -1 + 6(2) = 11$
• $a_3 = -1 + 6(3) = 17$
• $a_4 = -1 + 6(4) = 23$
• $a_5 = -1 + 6(5) = 29$

As we can see, the sequence generated by option C is $5, 11, 17, 23, 29, \ldots$, which is the same as the original sequence.

Final Conclusion

Based on our analysis, we can conclude that the explicit rule for the sequence $5, 11, 17, 23, 29, \ldots$ is $a_n = -1 + 6n$. This means that each term in the sequence is obtained by adding $6$ to the previous term, starting from the first term $a_1 = 5$.

Introduction

In our previous article, we discussed the explicit rule for the sequence $5, 11, 17, 23, 29, \ldots$. We analyzed the provided options and determined that the correct explicit rule is $a_n = -1 + 6n$. In this article, we will answer some frequently asked questions about the explicit rule and the sequence.

Q: What is the explicit rule for the sequence?

A: The explicit rule for the sequence $5, 11, 17, 23, 29, \ldots$ is $a_n = -1 + 6n$.

Q: How is the explicit rule used to generate the sequence?

A: The explicit rule is used to generate the sequence by substituting the term number $n$ into the equation $a_n = -1 + 6n$. For example, to generate the first term $a_1$, we substitute $n = 1$ into the equation:

$$a_1 = -1 + 6(1) = 5$$

Q: What is the common difference between consecutive terms in the sequence?

A: The common difference between consecutive terms in the sequence is $6$. This means that each term in the sequence is obtained by adding $6$ to the previous term.

Q: How can I use the explicit rule to find the next term in the sequence?

A: To find the next term in the sequence, substitute the current term number $n$ into the equation $a_n = -1 + 6n$, and then add $1$ to the result. For example, to find the next term after $a_5 = 29$, we substitute $n = 6$ into the equation:

$$a_6 = -1 + 6(6) = 35$$

Q: Can I use the explicit rule to find the nth term in the sequence?

A: Yes, you can use the explicit rule to find the nth term in the sequence by substituting $n$ into the equation $a_n = -1 + 6n$.

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

A: The formula for the nth term of the sequence is $a_n = -1 + 6n$.

Q: Can I use the explicit rule to find the sum of the first n terms of the sequence?

A: Yes, you can use the explicit rule to find the sum of the first n terms of the sequence by using the formula for the sum of an arithmetic series:

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

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

Q: Can I use the explicit rule to find the average of the first n terms of the sequence?

A: Yes, you can use the explicit rule to find the average of the first n terms of the sequence by using the formula:

$$\text{Average} = \frac{S_n}{n}$$

where $S_n$ is the sum of the first n terms.

Conclusion

In this article, we answered some frequently asked questions about the explicit rule for the sequence $5, 11, 17, 23, 29, \ldots$. We hope that this article has provided you with a better understanding of the explicit rule and how to use it to generate the sequence, find the next term, and calculate the sum and average of the first n terms.