The Ratio of the 11th and 8th Terms

Oct 13, 2025 by windailys.com 38 views

Introduction

Arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant. In this article, we will explore the concept of arithmetic progression and how to find the ratio of the sum of the first few terms. We will use the given information about the ratio of the 11th and 8th terms to find the ratio of the sum of the first 5 terms to the sum of the first 21 terms.

Arithmetic Progression

An arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant. The general form of an arithmetic progression is:

a, a + d, a + 2d, a + 3d, ...

where a is the first term and d is the common difference.

The Ratio of the 11th and 8th Terms

The problem states that the ratio of the 11th and 8th terms is 3:2. We can write this as:

(a + 10d) / (a + 7d) = 3/2

Solving for the Common Difference

To solve for the common difference, we can cross-multiply and simplify the equation:

2(a + 10d) = 3(a + 7d)

Expanding and simplifying the equation, we get:

2a + 20d = 3a + 21d

Subtracting 2a from both sides and subtracting 21d from both sides, we get:

d = a

Substituting this value of d into the original equation, we get:

(a + 10(-a)) / (a + 7(-a)) = 3/2

Simplifying the equation, we get:

(-9a) / (-3a) = 3/2

This equation is true for all values of a, so we can conclude that the common difference is:

d = -a

The Sum of the First n Terms

The sum of the first n terms of an arithmetic progression can be found using the formula:

S_n = n/2 [2a + (n-1)d]

The Ratio of the Sum of the First 5 Terms to the Sum of the First 21 Terms

We are asked to find the ratio of the sum of the first 5 terms to the sum of the first 21 terms. We can use the formula for the sum of the first n terms to find these values:

S_5 = 5/2 [2a + (5-1)d] = 5/2 [2a - 4a] = 5/2 [-2a]

S_21 = 21/2 [2a + (21-1)d] = 21/2 [2a - 20a] = 21/2 [-18a]

Finding the Ratio

We can now find the ratio of the sum of the first 5 terms to the sum of the first 21 terms:

S_5 / S_21 = (5/2 [-2a]) / (21/2 [-18a])

Simplifying the equation, we get:

S_5 / S_21 = (-5/2) / (-63/2)

Simplifying further, we get:

S_5 / S_21 = 5/63

Conclusion

In this article, we used the given information about the ratio of the 11th and 8th terms to find the ratio of the sum of the first 5 terms to the sum of the first 21 terms. We found that the ratio is 5:63.

References

[1] "Arithmetic Progression" by Khan Academy
[2] "Sum of an Arithmetic Series" by Math Open Reference

Introduction

In our previous article, we explored the concept of arithmetic progression and how to find the ratio of the sum of the first few terms. We used the given information about the ratio of the 11th and 8th terms to find the ratio of the sum of the first 5 terms to the sum of the first 21 terms. In this article, we will answer some frequently asked questions about the ratio of sums in arithmetic progression.

Q: What is an arithmetic progression?

A: An arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant. The general form of an arithmetic progression is:

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 ratio of the sum of the first n terms to the sum of the first m terms?

A: To find the ratio of the sum of the first n terms to the sum of the first m terms, you can use the formula for the sum of the first n terms:

S_n = n/2 [2a + (n-1)d]

S_m = m/2 [2a + (m-1)d]

Then, you can divide S_n by S_m to find the ratio.

Q: What is the formula for the sum of the first n terms of an arithmetic progression?

A: The formula for the sum of the first n terms of an arithmetic progression is:

S_n = n/2 [2a + (n-1)d]

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

A: To find the common difference in an arithmetic progression, you can use the formula:

d = (a_n - a_1) / (n - 1)

where a_n is the nth term, a_1 is the first term, and n is the number of terms.

Q: What is the ratio of the sum of the first 5 terms to the sum of the first 21 terms in an arithmetic progression?

A: In our previous article, we found that the ratio of the sum of the first 5 terms to the sum of the first 21 terms is 5:63.

Q: Can I use the formula for the sum of the first n terms to find the sum of the first m terms?

A: Yes, you can use the formula for the sum of the first n terms to find the sum of the first m terms. However, you need to make sure that the formula is valid for the given values of n and m.

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

A: Arithmetic progression has many real-world applications, including:

Finance: The interest on a loan or investment can be calculated using arithmetic progression.
Music: The frequency of notes in a musical scale can be calculated using arithmetic progression.
Science: The motion of an object under constant acceleration can be calculated using arithmetic progression.

Conclusion

In this article, we answered some frequently asked questions about the ratio of sums in arithmetic progression. We hope that this article has been helpful in understanding the concept of arithmetic progression and how to find the ratio of the sum of the first few terms.

References

[1] "Arithmetic Progression" by Khan Academy
[2] "Sum of an Arithmetic Series" by Math Open Reference