Recalling the Formula for the nth Term of an AP

Oct 13, 2025 by windailys.com 73 views

Introduction

Arithmetic Progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant. In this article, we will explore how to find the last three terms of an arithmetic progression given some of its terms. We will use the information provided about the 4th, 8th, 6th, and 10th terms to solve the problem.

Understanding the Problem

We are given an arithmetic progression with 30 terms. Some of the terms are:

The 4th term is 24.
The 8th term is 24.
The sum of the 6th and 10th terms is 44.

We need to find the last three terms of this arithmetic progression.

Recalling the Formula for the nth Term of an AP

The formula for the nth term of an arithmetic progression is:

an = a1 + (n - 1)d

where an is the nth term, a1 is the first term, n is the term number, and d is the common difference.

Finding the Common Difference

We are given that the 4th term is 24 and the 8th term is 24. We can use this information to find the common difference.

Let's denote the first term as a and the common difference as d. Then, we can write the 4th term as:

a + 3d = 24

We can also write the 8th term as:

a + 7d = 24

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

4d = 0

This implies that d = 0. However, this is not possible since the terms are not constant. Let's re-examine the problem.

Re-examining the Problem

We are given that the 4th term is 24 and the 8th term is 24. This means that the 4th and 8th terms are equal. In an arithmetic progression, this is only possible if the common difference is zero. However, this is not the case here.

Let's try to find the common difference using a different approach.

Using the Sum of the 6th and 10th Terms

We are given that the sum of the 6th and 10th terms is 44. We can write this as:

a + 5d + a + 9d = 44

Combine like terms:

2a + 14d = 44

Finding the Common Difference

We can use the information provided about the 4th and 8th terms to find the common difference. Let's denote the first term as a and the common difference as d. Then, we can write the 4th term as:

a + 3d = 24

We can also write the 8th term as:

a + 7d = 24

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

4d = 0

This implies that d = 0. However, this is not possible since the terms are not constant.

Let's try to find the common difference using a different approach.

Using the Sum of the 6th and 10th Terms

We are given that the sum of the 6th and 10th terms is 44. We can write this as:

a + 5d + a + 9d = 44

Combine like terms:

2a + 14d = 44

We can also write the 4th term as:

a + 3d = 24

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

11d = 20

This implies that d = 20/11.

Finding the First Term

We can use the information provided about the 4th term to find the first term. Let's denote the first term as a and the common difference as d. Then, we can write the 4th term as:

a + 3d = 24

Substituting d = 20/11, we get:

a + 3(20/11) = 24

Simplifying, we get:

a + 60/11 = 24

Subtracting 60/11 from both sides, we get:

a = 24 - 60/11

Simplifying, we get:

a = (264 - 60)/11

a = 204/11

Finding the Last Three Terms

We can use the formula for the nth term of an arithmetic progression to find the last three terms.

The 29th term is:

a + 28d = 204/11 + 28(20/11)

Simplifying, we get:

a + 28d = 204/11 + 560/11

a + 28d = 764/11

The 30th term is:

a + 29d = 204/11 + 29(20/11)

Simplifying, we get:

a + 29d = 204/11 + 580/11

a + 29d = 784/11

The 31st term is:

a + 30d = 204/11 + 30(20/11)

Simplifying, we get:

a + 30d = 204/11 + 600/11

a + 30d = 804/11

Conclusion

In this article, we used the information provided about the 4th, 8th, 6th, and 10th terms to find the last three terms of an arithmetic progression. We found that the common difference is 20/11 and the first term is 204/11. We then used the formula for the nth term of an arithmetic progression to find the last three terms.

Final Answer

The last three terms of the arithmetic progression are:

The 29th term is 764/11.
The 30th term is 784/11.
The 31st term is 804/11.
Frequently Asked Questions (FAQs) =====================================

Q: What is an arithmetic progression?

A: An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant.

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

A: To find the common difference, you can use the formula for the nth term of an arithmetic progression: an = a1 + (n - 1)d. You can also use the information provided about the terms to find the common difference.

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

A: The formula for the nth term of an arithmetic progression is: an = a1 + (n - 1)d, where an is the nth term, a1 is the first term, n is the term number, and d is the common difference.

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

A: To find the first term, you can use the information provided about the terms and the formula for the nth term of an arithmetic progression.

Q: What is the significance of the common difference in an arithmetic progression?

A: The common difference is the difference between any two consecutive terms in an arithmetic progression. It is a key component in finding the terms of the progression.

Q: Can I use the information provided about the terms to find the common difference?

A: Yes, you can use the information provided about the terms to find the common difference. For example, if you know the 4th and 8th terms are equal, you can use this information to find the common difference.

Q: How do I find the last three terms of an arithmetic progression?

A: To find the last three terms, you can use the formula for the nth term of an arithmetic progression and the information provided about the terms.

Q: What if I don't have enough information to find the common difference?

A: If you don't have enough information to find the common difference, you may not be able to find the terms of the arithmetic progression.

Q: Can I use a calculator to find the terms of an arithmetic progression?

A: Yes, you can use a calculator to find the terms of an arithmetic progression. However, it's always a good idea to understand the underlying math and formulas.

Q: How do I know if a sequence is an arithmetic progression?

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

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

A: Arithmetic progressions have many real-world applications, such as finance, music, and science. For example, the interest on a loan or the frequency of notes in music can be represented as an arithmetic progression.

Q: Can I use arithmetic progressions to model real-world phenomena?

A: Yes, arithmetic progressions can be used to model real-world phenomena, such as population growth, temperature changes, and financial transactions.

Q: How do I extend an arithmetic progression to find more terms?

A: To extend an arithmetic progression, you can use the formula for the nth term of an arithmetic progression and the information provided about the terms.

Q: What are some common mistakes to avoid when working with arithmetic progressions?

A: Some common mistakes to avoid when working with arithmetic progressions include:

Assuming the common difference is zero without checking.
Not using the correct formula for the nth term of an arithmetic progression.
Not checking if the sequence is an arithmetic progression before trying to find the terms.

Q: How do I check if a sequence is an arithmetic progression?

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