The Sum Of The First Three Terms Of An Arithmetic Sequence Is 21. If The Sum Of The First Two Terms Is Subtracted From The Third Term, It Results In 9. Find The Three Terms Of The Series.

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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 explore the problem of finding the three terms of an arithmetic sequence given the sum of the first three terms and the result of subtracting the sum of the first two terms from the third term.

Problem Statement

Let's denote the first term of the arithmetic sequence as a, the second term as a + d, and the third term as a + 2d, where d is the common difference between the terms. We are given that the sum of the first three terms is 21, which can be expressed as:

a + (a + d) + (a + 2d) = 21

We are also given that the result of subtracting the sum of the first two terms from the third term is 9, which can be expressed as:

(a + 2d) - (a + d) = 9

Solving the System of Equations

We can simplify the first equation by combining like terms:

3a + 3d = 21

Dividing both sides by 3, we get:

a + d = 7

Now, let's simplify the second equation:

a + d = 9

Subtracting the first equation from the second equation, we get:

0 = 2

This is a contradiction, which means that the system of equations has no solution. However, we can try to find a solution by assuming that the common difference d is not equal to 0.

Assuming a Non-Zero Common Difference

Let's assume that d is not equal to 0. Then, we can rewrite the second equation as:

a + d = 9

Subtracting the first equation from the second equation, we get:

2d = 2

Dividing both sides by 2, we get:

d = 1

Now, let's substitute d = 1 into the first equation:

a + 1 = 7

Subtracting 1 from both sides, we get:

a = 6

Now that we have found the values of a and d, we can find the three terms of the arithmetic sequence:

a = 6 a + d = 6 + 1 = 7 a + 2d = 6 + 2(1) = 8

Conclusion

In this article, we have explored the problem of finding the three terms of an arithmetic sequence given the sum of the first three terms and the result of subtracting the sum of the first two terms from the third term. We have assumed that the common difference d is not equal to 0 and have found the values of a and d. We have then used these values to find the three terms of the arithmetic sequence.

The Three Terms of the Arithmetic Sequence

The three terms of the arithmetic sequence are:

  • a = 6
  • a + d = 6 + 1 = 7
  • a + 2d = 6 + 2(1) = 8

Arithmetic Sequences and Their Applications

Arithmetic sequences have numerous applications in various fields, including finance, engineering, and science. They are used to model real-world phenomena, such as population growth, financial investments, and physical systems. In finance, arithmetic sequences are used to calculate compound interest and to model the growth of investments. In engineering, arithmetic sequences are used to model the behavior of physical systems, such as the motion of objects and the flow of fluids.

Real-World Applications of Arithmetic Sequences

Arithmetic sequences have numerous real-world applications, including:

  • Finance: Arithmetic sequences are used to calculate compound interest and to model the growth of investments.
  • Engineering: Arithmetic sequences are used to model the behavior of physical systems, such as the motion of objects and the flow of fluids.
  • Science: Arithmetic sequences are used to model real-world phenomena, such as population growth and the behavior of physical systems.

Conclusion

Introduction

Arithmetic sequences are a fundamental concept in mathematics, and they have numerous applications in various fields, including finance, engineering, and science. In this article, we will provide a Q&A guide to help you understand arithmetic sequences and their applications.

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. For example, the sequence 2, 4, 6, 8, 10 is an arithmetic sequence with a common difference of 2.

Q: What are the characteristics of an arithmetic sequence?

A: The characteristics of an arithmetic sequence are:

  • The difference between any two consecutive terms is constant.
  • The sequence can be represented by the formula: a, a + d, a + 2d, a + 3d, ...
  • Where a is the first term and d is the common difference.

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

A: To find the common difference of an arithmetic sequence, you can use the following formula:

d = (term_n - term_(n-1))

Where d is the common difference, term_n is the nth term, and term_(n-1) is the (n-1)th term.

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 following formula:

term_n = a + (n-1)d

Where term_n is the nth term, a is the first term, n is the term number, and d is the common difference.

Q: What are the applications of arithmetic sequences?

A: Arithmetic sequences have numerous applications in various fields, including:

  • Finance: Arithmetic sequences are used to calculate compound interest and to model the growth of investments.
  • Engineering: Arithmetic sequences are used to model the behavior of physical systems, such as the motion of objects and the flow of fluids.
  • Science: Arithmetic sequences are used to model real-world phenomena, such as population growth and the behavior of physical systems.

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

A: To find the sum of an arithmetic sequence, you can use the following formula:

S_n = (n/2)(a + term_n)

Where S_n is the sum of the first n terms, n is the term number, a is the first term, and term_n is the nth term.

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

A: The formula for the sum of an infinite arithmetic sequence is:

S = a / (1 - r)

Where S is the sum of the infinite sequence, a is the first term, and r is the common ratio.

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

A: The formula for the sum of a finite arithmetic sequence is:

S_n = (n/2)(a + term_n)

Where S_n is the sum of the first n terms, n is the term number, a is the first term, and term_n is the nth term.

Conclusion

In conclusion, arithmetic sequences are a fundamental concept in mathematics, and they have numerous applications in various fields, including finance, engineering, and science. In this article, we have provided a Q&A guide to help you understand arithmetic sequences and their applications. We hope that this guide has been helpful in answering your questions and providing you with a better understanding of arithmetic sequences.