The nth Term of the Linear Sequence

Oct 13, 2025 by windailys.com 2 views

The nth Term of a Linear Sequence: Understanding the 25th Term and the Difference with Another Arithmetic Sequence

In mathematics, a linear sequence is a sequence of numbers in which each term after the first is obtained by adding a fixed constant to the previous term. The nth term of a linear sequence can be represented by the formula $an = a_1 + (n - 1)d$ , where $a_n$ is the nth term, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference. In this article, we will explore the nth term of a linear sequence given by the formula $4n - 5$ and work out the 25th term of the sequence. Additionally, we will examine another arithmetic sequence with the nth term $8n - 16$ and discuss why John is wrong in saying that there is a number that is in both sequences.

The nth Term of the Linear Sequence

The nth term of the linear sequence is given by the formula $4n - 5$ . To understand this formula, let's break it down into its components. The term $4n$ represents the product of the coefficient $4$ and the term number $n$ . The term $-5$ is a constant that is subtracted from the product $4n$ . This means that each term in the sequence is obtained by multiplying the term number by $4$ and then subtracting $5$ .

Working Out the 25th Term of the Sequence

To work out the 25th term of the sequence, we can simply substitute $n = 25$ into the formula $4n - 5$ . This gives us:

$4(25) - 5 = 100 - 5 = 95$

Therefore, the 25th term of the sequence is $95$ .

Another Arithmetic Sequence

Another arithmetic sequence has the nth term $8n - 16$ . This sequence can be represented by the formula $8n - 16$ , where $8$ is the common difference and $-16$ is the first term. To understand this formula, let's break it down into its components. The term $8n$ represents the product of the coefficient $8$ and the term number $n$ . The term $-16$ is a constant that is subtracted from the product $8n$ . This means that each term in the sequence is obtained by multiplying the term number by $8$ and then subtracting $16$ .

Why John is Wrong

John says that there is a number that is in both sequences. However, this is not true. The two sequences have different common differences and first terms, which means that they are not the same sequence. The first sequence has a common difference of $4$ and a first term of $-5$ , while the second sequence has a common difference of $8$ and a first term of $-16$ . Since the two sequences have different common differences and first terms, there is no number that is in both sequences.

In conclusion, the nth term of a linear sequence is given by the formula $4n - 5$ . We worked out the 25th term of the sequence and found that it is $95$ . We also examined another arithmetic sequence with the nth term $8n - 16$ and discussed why John is wrong in saying that there is a number that is in both sequences. The two sequences have different common differences and first terms, which means that they are not the same sequence.

The nth term of a linear sequence can be represented by the formula $an = a_1 + (n - 1)d$ .
The nth term of the linear sequence $4n - 5$ can be obtained by multiplying the term number by $4$ and then subtracting $5$ .
The 25th term of the sequence is $95$ .
The two sequences have different common differences and first terms, which means that they are not the same sequence.

For further reading on linear sequences and arithmetic sequences, we recommend the following resources:

[1] Khan Academy. (n.d.). Linear Sequences. Retrieved from https://www.khanacademy.org/math/algebra/linear-equations-and-functions/linear-sequences/v/linear-sequences
[2] Math Open Reference. (n.d.). Arithmetic Sequences. Retrieved from https://www.mathopenref.com/arithmeticsequences.html
Q&A: The nth Term of a Linear Sequence and Arithmetic Sequences

In our previous article, we explored the nth term of a linear sequence given by the formula $4n - 5$ and worked out the 25th term of the sequence. We also examined another arithmetic sequence with the nth term $8n - 16$ and discussed why John is wrong in saying that there is a number that is in both sequences. In this article, we will answer some frequently asked questions about linear sequences and arithmetic sequences.

Q: What is the difference between a linear sequence and an arithmetic sequence?

A: A linear sequence is a sequence of numbers in which each term after the first is obtained by adding a fixed constant to the previous term. An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a fixed constant to the previous term, and the common difference is the same between each pair of consecutive terms.

Q: How do I determine the nth term of a linear sequence?

A: To determine the nth term of a linear sequence, you can use the formula $an = a_1 + (n - 1)d$ , where $a_n$ is the nth term, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference.

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

A: The formula for the nth term of an arithmetic sequence is $an = a_1 + (n - 1)d$ , where $a_n$ is the nth term, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference.

Q: How do I find the 25th term of a linear sequence?

A: To find the 25th term of a linear sequence, you can substitute $n = 25$ into the formula $an = a_1 + (n - 1)d$ .

Q: Why is it important to understand linear sequences and arithmetic sequences?

A: Understanding linear sequences and arithmetic sequences is important because they are used in many real-world applications, such as finance, economics, and science. They are also used in computer programming and data analysis.

Q: Can you give an example of a linear sequence in real life?

A: Yes, a linear sequence can be seen in the growth of a population over time. For example, if a population grows at a rate of 2% per year, the number of people in the population can be represented by a linear sequence.

Q: Can you give an example of an arithmetic sequence in real life?

A: Yes, an arithmetic sequence can be seen in the cost of a product over time. For example, if the cost of a product increases by $5 each year, the cost of the product can be represented by an arithmetic sequence.

Q: How do I determine if a sequence is linear or arithmetic?

A: To determine if a sequence is linear or arithmetic, you can look at the pattern of the sequence. If the sequence has a constant difference between each pair of consecutive terms, it is an arithmetic sequence. If the sequence has a constant difference between each pair of consecutive terms, but the difference is not the same between each pair of consecutive terms, it is a linear sequence.

In conclusion, understanding linear sequences and arithmetic sequences is important for many real-world applications. By knowing the formulas for the nth term of a linear sequence and an arithmetic sequence, you can determine the nth term of a sequence and understand the pattern of the sequence. We hope that this article has helped you to understand linear sequences and arithmetic sequences better.

A linear sequence is a sequence of numbers in which each term after the first is obtained by adding a fixed constant to the previous term.
An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a fixed constant to the previous term, and the common difference is the same between each pair of consecutive terms.
The formula for the nth term of a linear sequence is $an = a_1 + (n - 1)d$ .
The formula for the nth term of an arithmetic sequence is $an = a_1 + (n - 1)d$ .
Understanding linear sequences and arithmetic sequences is important for many real-world applications.

For further reading on linear sequences and arithmetic sequences, we recommend the following resources:

[1] Khan Academy. (n.d.). Linear Sequences. Retrieved from https://www.khanacademy.org/math/algebra/linear-equations-and-functions/linear-sequences/v/linear-sequences
[2] Math Open Reference. (n.d.). Arithmetic Sequences. Retrieved from https://www.mathopenref.com/arithmeticsequences.html