The Arithmetic Sequence Formula

Oct 13, 2025 by leolavanderia.com.br 17 views

Introduction

Arithmetic sequences are a fundamental concept in mathematics, and they have numerous applications in various fields, including finance, engineering, and science. An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. In this article, we will use the arithmetic sequence formula to find the missing values in a given table.

The Arithmetic Sequence Formula

The arithmetic sequence formula is given by:

a_n = a_1 + (n-1)d

where:

$a_n$ is the nth term of the sequence
$a_1$ is the first term of the sequence
$n$ is the term number
$d$ is the common difference between consecutive terms

However, in this problem, we will use the formula $a_n = -8 + (n-1) \cdot 2$ to find the missing values.

Step 1: Understanding the Formula

Let's break down the formula $a_n = -8 + (n-1) \cdot 2$ . This formula tells us that each term in the sequence is obtained by adding 2 to the previous term, starting from -8.

Step 2: Finding the Missing Values

We are given a table with four options, and we need to find the missing values using the formula $a_n = -8 + (n-1) \cdot 2$ . Let's analyze each option:

Option A: $a = -12 ; b = -10$

To find the missing values, we can plug in the values of $a$ and $b$ into the formula. Let's start with $a = -12$ . We can set $n = 1$ and solve for $a_1$ :

-12 = a_1 + (1-1) \cdot 2

Simplifying the equation, we get:

-12 = a_1

So, the first term of the sequence is $a_1 = -12$ . Now, let's find the second term $b$ :

-10 = -8 + (2-1) \cdot 2

Simplifying the equation, we get:

-10 = -8 + 2

-10 = -6

This is not correct, so option A is not the correct answer.

Option B: $a = -15 ; b = -13$

Let's plug in the values of $a$ and $b$ into the formula. We can start with $a = -15$ . We can set $n = 1$ and solve for $a_1$ :

-15 = a_1 + (1-1) \cdot 2

Simplifying the equation, we get:

-15 = a_1

So, the first term of the sequence is $a_1 = -15$ . Now, let's find the second term $b$ :

-13 = -8 + (2-1) \cdot 2

Simplifying the equation, we get:

-13 = -8 + 2

-13 = -6

This is not correct, so option B is not the correct answer.

Option C: $a = -6 ; b = -2$

Let's plug in the values of $a$ and $b$ into the formula. We can start with $a = -6$ . We can set $n = 1$ and solve for $a_1$ :

-6 = a_1 + (1-1) \cdot 2

Simplifying the equation, we get:

-6 = a_1

So, the first term of the sequence is $a_1 = -6$ . Now, let's find the second term $b$ :

-2 = -8 + (2-1) \cdot 2

Simplifying the equation, we get:

-2 = -8 + 2

-2 = -6

This is not correct, so option C is not the correct answer.

Option D: $a = -10 ; b = -8$

Let's plug in the values of $a$ and $b$ into the formula. We can start with $a = -10$ . We can set $n = 1$ and solve for $a_1$ :

-10 = a_1 + (1-1) \cdot 2

Simplifying the equation, we get:

-10 = a_1

So, the first term of the sequence is $a_1 = -10$ . Now, let's find the second term $b$ :

-8 = -8 + (2-1) \cdot 2

Simplifying the equation, we get:

-8 = -8

This is correct, so option D is the correct answer.

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 is the formula for an arithmetic sequence?

A: The formula for an arithmetic sequence is given by:

a_n = a_1 + (n-1)d

where:

$a_n$ is the nth term of the sequence
$a_1$ is the first term of the sequence
$n$ is the term number
$d$ is the common difference between consecutive terms

Q: How do I find the missing values in an arithmetic sequence?

A: To find the missing values in an arithmetic sequence, you can use the formula $a_n = a_1 + (n-1)d$ . Plug in the values of $a_1$ , $n$ , and $d$ to find the missing value.

Q: What is the common difference in an arithmetic sequence?

A: The common difference in an arithmetic sequence is the difference between any two consecutive terms. It is denoted by the letter $d$ .

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

A: To find the common difference in an arithmetic sequence, you can subtract any two consecutive terms. For example, if the sequence is $a, b, c, d$ , then the common difference is $b - a$ .

Q: What is the first term of an arithmetic sequence?

A: The first term of an arithmetic sequence is the first number in the sequence. It is denoted by the letter $a_1$ .

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

A: To find the first term of an arithmetic sequence, you can use the formula $a_n = a_1 + (n-1)d$ . Plug in the values of $a_n$ , $n$ , and $d$ to find the first term.

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

A: The nth term of an arithmetic sequence is the nth number in the sequence. It is denoted by the letter $a_n$ .

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 $a_n = a_1 + (n-1)d$ . Plug in the values of $a_1$ , $n$ , and $d$ to find the nth term.

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

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

Finance: Arithmetic sequences are used to calculate interest rates and investment returns.
Engineering: Arithmetic sequences are used to design and optimize systems, such as bridges and buildings.
Science: Arithmetic sequences are used to model population growth and decay.
Music: Arithmetic sequences are used to create musical patterns and rhythms.

Q: How do I use arithmetic sequences in real-world applications?

A: To use arithmetic sequences in real-world applications, you can use the formula $a_n = a_1 + (n-1)d$ . Plug in the values of $a_1$ , $n$ , and $d$ to find the missing value. Then, use the arithmetic sequence to model and analyze the real-world situation.

Conclusion

In this article, we answered some frequently asked questions about arithmetic sequences. We covered topics such as the formula for an arithmetic sequence, finding missing values, and real-world applications. We hope this article has been helpful in understanding arithmetic sequences and their applications.