Which Table Represents An Arithmetic Sequence?${ \begin{tabular}{|c|c|c|c|c|c|} \hline N N N & 1 & 2 & 3 & 4 & 5 \ \hline A N A _{ N } A N & -6 & -12 & -24 & -48 & -96 \ \hline \end{tabular} }$[ \begin{tabular}{|c|c|c|c|c|c|} \hline N N N & 1

Mar 11, 2025 by ADMIN 247 views

**Which Table Represents an Arithmetic Sequence?**

Introduction

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. In other words, if we have a sequence of numbers, and the difference between the first and second term is the same as the difference between the second and third term, and so on, then that sequence is an arithmetic sequence. In this article, we will explore which table represents an arithmetic sequence.

Understanding Arithmetic Sequences

Before we dive into the tables, let's understand what makes an arithmetic sequence. An arithmetic sequence has the following properties:

The difference between any two consecutive terms is constant.
The sequence can be represented by the formula: $a_n = a_1 + (n-1)d$ , where $a_n$ is the nth term, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference.

Table 1:

Let's examine the first table:

| $n$ | 1 | 2 | 3 | 4 | 5 | | --- | -6 | -12 | -24 | -48 | -96 |

In this table, the difference between any two consecutive terms is not constant. For example, the difference between the first and second term is -6, while the difference between the second and third term is -12, which is twice the difference between the first and second term. This means that the sequence in Table 1 is not an arithmetic sequence.

Table 2:

Now, let's examine the second table:

| $n$ | 1 | 2 | 3 | 4 | 5 | | --- | 2 | 5 | 8 | 11 | 14 |

In this table, the difference between any two consecutive terms is constant. For example, the difference between the first and second term is 3, while the difference between the second and third term is also 3. This means that the sequence in Table 2 is an arithmetic sequence.

Conclusion

In conclusion, the table that represents an arithmetic sequence is Table 2. The difference between any two consecutive terms in Table 2 is constant, which is a defining characteristic of an arithmetic sequence. On the other hand, the sequence in Table 1 is not an arithmetic sequence because the difference between any two consecutive terms is not constant.

Arithmetic Sequences in Real-Life Applications

Arithmetic sequences have numerous real-life applications. For example, in finance, an arithmetic sequence can be used to model the growth of an investment over time. In music, an arithmetic sequence can be used to create a musical scale. In sports, an arithmetic sequence can be used to track the progress of an athlete over time.

Examples of Arithmetic Sequences

Here are a few examples of arithmetic sequences:

The sequence 2, 5, 8, 11, 14 is an arithmetic sequence with a common difference of 3.
The sequence -6, -12, -24, -48, -96 is not an arithmetic sequence because the difference between any two consecutive terms is not constant.
The sequence 1, 4, 7, 10, 13 is an arithmetic sequence with a common difference of 3.

Properties of Arithmetic Sequences

Arithmetic sequences have several important properties, including:

The sum of an arithmetic sequence is given by the formula: $S_n = \frac{n}{2}(a_1 + a_n)$ , where $S_n$ is the sum of the first n terms, $a_1$ is the first term, $a_n$ is the nth term, and $n$ is the number of terms.
The average of an arithmetic sequence is given by the formula: $A_n = \frac{a_1 + a_n}{2}$ , where $A_n$ is the average of the first n terms, $a_1$ is the first term, and $a_n$ is the nth term.

Solved Problems

Here are a few solved problems involving arithmetic sequences:

Find the sum of the first 5 terms of the arithmetic sequence 2, 5, 8, 11, 14.
Find the average of the first 5 terms of the arithmetic sequence 2, 5, 8, 11, 14.
Determine whether the sequence -6, -12, -24, -48, -96 is an arithmetic sequence.

Answer Key

Here are the answers to the solved problems:

The sum of the first 5 terms of the arithmetic sequence 2, 5, 8, 11, 14 is 40.
The average of the first 5 terms of the arithmetic sequence 2, 5, 8, 11, 14 is 8.
The sequence -6, -12, -24, -48, -96 is not an arithmetic sequence because the difference between any two consecutive terms is not constant.

Conclusion

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.

Q: What are the properties of an arithmetic sequence?

A: The properties of an arithmetic sequence include:

The difference between any two consecutive terms is constant.
The sequence can be represented by the formula: $a_n = a_1 + (n-1)d$ , 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 determine if a sequence is an arithmetic sequence?

A: To determine if a sequence is an arithmetic sequence, you can check if the difference between any two consecutive terms is constant. If it is, then the sequence is an arithmetic sequence.

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

A: The formula for the sum of an arithmetic sequence is: $S_n = \frac{n}{2}(a_1 + a_n)$ , where $S_n$ is the sum of the first n terms, $a_1$ is the first term, $a_n$ is the nth term, and $n$ is the number of terms.

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

A: The formula for the average of an arithmetic sequence is: $A_n = \frac{a_1 + a_n}{2}$ , where $A_n$ is the average of the first n terms, $a_1$ is the first term, and $a_n$ is the nth term.

Q: Can an arithmetic sequence have a negative common difference?

A: Yes, an arithmetic sequence can have a negative common difference. For example, the sequence -6, -12, -24, -48, -96 has a negative common difference of 6.

Q: Can an arithmetic sequence have a zero common difference?

A: Yes, an arithmetic sequence can have a zero common difference. For example, the sequence 1, 1, 1, 1, 1 has a zero common difference.

Q: What is the relationship between arithmetic sequences and geometric sequences?

A: Arithmetic sequences and geometric sequences are two different types of sequences. Arithmetic sequences have a constant difference between terms, while geometric sequences have a constant ratio between terms.

Q: Can an arithmetic sequence be a geometric sequence?

A: No, an arithmetic sequence cannot be a geometric sequence. The two types of sequences have different properties and characteristics.

Q: What are some real-life applications of arithmetic sequences?

A: Arithmetic sequences have numerous real-life applications, including:

Finance: Arithmetic sequences can be used to model the growth of an investment over time.
Music: Arithmetic sequences can be used to create a musical scale.
Sports: Arithmetic sequences can be used to track the progress of an athlete over time.

Q: How do I find the nth term of an arithmetic sequence?

A: To find the nth term of an arithmetic sequence, you can use the formula: $a_n = a_1 + (n-1)d$ , 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 sum of the first n terms of an arithmetic sequence?

A: To find the sum of the first n terms of an arithmetic sequence, you can use the formula: $S_n = \frac{n}{2}(a_1 + a_n)$ , where $S_n$ is the sum of the first n terms, $a_1$ is the first term, $a_n$ is the nth term, and $n$ is the number of terms.

Q: How do I find the average of the first n terms of an arithmetic sequence?

A: To find the average of the first n terms of an arithmetic sequence, you can use the formula: $A_n = \frac{a_1 + a_n}{2}$ , where $A_n$ is the average of the first n terms, $a_1$ is the first term, and $a_n$ is the nth term.