What Are The Next 4 Terms Of The Sequence 1 , 6 , 11 , … 1, 6, 11, \ldots 1 , 6 , 11 , … ?A) 15 , 19 , 23 , 27 15, 19, 23, 27 15 , 19 , 23 , 27 B) 16 , 21 , 26 , 31 16, 21, 26, 31 16 , 21 , 26 , 31 C) 18 , 26 , 35 , 45 18, 26, 35, 45 18 , 26 , 35 , 45 D) 6 , 1 , − 4 , − 9 6, 1, -4, -9 6 , 1 , − 4 , − 9

Introduction

In mathematics, sequences are an essential concept that helps us understand patterns and relationships between numbers. A sequence is a list of numbers in a specific order, and each number is called a term. In this article, we will delve into the world of sequences and explore the next four terms of the given sequence $1, 6, 11, \ldots$. We will analyze the pattern, identify the common difference, and use it to determine the next four terms.

Understanding the Sequence

The given sequence is $1, 6, 11, \ldots$. To understand the pattern, let's examine the differences between consecutive terms:

• $6 - 1 = 5$
• $11 - 6 = 5$

We can see that the difference between consecutive terms is constant, which is $5$. This indicates that the sequence is an arithmetic sequence with a common difference of $5$.

Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. The general formula for an arithmetic sequence is:

$a_n = a_1 + (n - 1)d$

where $a_n$ is the $n$th term, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference.

In our sequence, $a_1 = 1$ and $d = 5$. We can use this formula to find the next four terms.

Finding the Next Four Terms

To find the next four terms, we will use the formula for an arithmetic sequence:

$a_n = a_1 + (n - 1)d$

We know that $a_1 = 1$ and $d = 5$. We want to find the next four terms, so we will substitute $n = 4, 5, 6, 7$ into the formula:

• For $n = 4$: $a_4 = 1 + (4 - 1)5$ $a_4 = 1 + 15$ $a_4 = 16$

• For $n = 5$: $a_5 = 1 + (5 - 1)5$ $a_5 = 1 + 20$ $a_5 = 21$

• For $n = 6$: $a_6 = 1 + (6 - 1)5$ $a_6 = 1 + 25$ $a_6 = 26$

• For $n = 7$: $a_7 = 1 + (7 - 1)5$ $a_7 = 1 + 30$ $a_7 = 31$

Therefore, the next four terms of the sequence are $16, 21, 26, 31$.

Conclusion

In this article, we explored the sequence $1, 6, 11, \ldots$ and found the next four terms using the formula for an arithmetic sequence. We identified the common difference as $5$ and used it to determine the next four terms. The correct answer is $16, 21, 26, 31$.

Answer

The correct answer is:

• B) $16, 21, 26, 31$

Discussion

Introduction

In our previous article, we explored the sequence $1, 6, 11, \ldots$ and found the next four terms using the formula for an arithmetic sequence. We identified the common difference as $5$ and used it to determine the next four terms. In this article, we will answer some frequently asked questions related to the sequence and provide additional insights.

Q&A

Q: What is an arithmetic sequence?

A: An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. The general formula for an arithmetic sequence is:

$a_n = a_1 + (n - 1)d$

where $a_n$ is the $n$th term, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference.

Q: How do I identify the common difference in an arithmetic sequence?

A: To identify the common difference, you can subtract any two consecutive terms. For example, in the sequence $1, 6, 11, \ldots$, you can subtract the first term from the second term:

$6 - 1 = 5$

This indicates that the common difference is $5$.

Q: How do I find the next term in an arithmetic sequence?

A: To find the next term in an arithmetic sequence, you can use the formula:

$a_n = a_1 + (n - 1)d$

where $a_n$ is the $n$th term, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference.

For example, in the sequence $1, 6, 11, \ldots$, you can find the next term by substituting $n = 4$ into the formula:

$a_4 = 1 + (4 - 1)5$ $a_4 = 1 + 15$ $a_4 = 16$

Q: What is the formula for an arithmetic sequence?

A: The formula for an arithmetic sequence is:

$a_n = a_1 + (n - 1)d$

where $a_n$ is the $n$th term, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference.

Q: Can I use the formula to find the first term of an arithmetic sequence?

A: Yes, you can use the formula to find the first term of an arithmetic sequence. If you know the $n$th term, the common difference, and the term number, you can substitute these values into the formula to find the first term.

For example, if you know that the $4$th term is $16$, the common difference is $5$, and the term number is $4$, you can substitute these values into the formula to find the first term:

$16 = a_1 + (4 - 1)5$ $16 = a_1 + 15$ $a_1 = 1$

Q: Can I use the formula to find the common difference of an arithmetic sequence?

A: Yes, you can use the formula to find the common difference of an arithmetic sequence. If you know the $n$th term, the first term, and the term number, you can substitute these values into the formula to find the common difference.

For example, if you know that the $4$th term is $16$, the first term is $1$, and the term number is $4$, you can substitute these values into the formula to find the common difference:

$16 = 1 + (4 - 1)5$ $16 = 1 + 15$ $5 = 5$

Conclusion

In this article, we answered some frequently asked questions related to the sequence and provided additional insights. We hope that this article has been helpful in understanding arithmetic sequences and how to use the formula to find the next terms. If you have any further questions or would like to discuss this topic further, please feel free to ask.

Additional Resources

• Arithmetic Sequences and Series
• Arithmetic Sequence Formula

Discussion

This problem requires a strong understanding of arithmetic sequences and the ability to apply the formula to find the next terms. It also requires attention to detail and the ability to identify the common difference. If you have any questions or would like to discuss this problem further, please feel free to ask.