Understanding Infinite Geometric Progression

Oct 13, 2025 by magicalceylontours.com 4 views

Introduction

In mathematics, an infinite geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The sum of an infinite geometric progression can be calculated using a simple formula, but the progression itself can be quite complex. In this article, we will explore how to determine the progression given the sum of the first three terms and the sum of the entire progression.

Understanding Infinite Geometric Progression

An infinite geometric progression is a sequence of numbers that can be represented as:

a, ar, ar^2, ar^3, ...

where 'a' is the first term and 'r' is the common ratio. The sum of the first three terms is given by:

a + ar + ar^2 = (1 + r + r^2)a

The sum of the entire progression is given by:

S = a / (1 - r)

Given Information

We are given that the sum of the first three terms is 26/3, and the sum of the entire progression is 9. We can represent this information mathematically as:

(1 + r + r^2)a = 26/3

a / (1 - r) = 9

Solving for the Common Ratio

We can start by solving the second equation for 'a':

a = 9(1 - r)

Now, substitute this expression for 'a' into the first equation:

(1 + r + r^2)(9(1 - r)) = 26/3

Expand and simplify the equation:

9(1 + r + r^2 - r - r^2) = 26/3

9(1 - r) = 26/3

9 = 26/3(1 - r)

27 = 26(1 - r)

27 = 26 - 26r

26r = 26 - 27

26r = -1

r = -1/26

Solving for the First Term

Now that we have the common ratio, we can substitute it back into the expression for 'a':

a = 9(1 - r)

a = 9(1 - (-1/26))

a = 9(1 + 1/26)

a = 9(27/26)

a = 243/26

Determining the Progression

Now that we have the first term and the common ratio, we can determine the progression. The progression is given by:

a, ar, ar^2, ar^3, ...

Substitute the values of 'a' and 'r' into the progression:

243/26, (243/26)(-1/26), ((243/26)(-1/26))^2, ((243/26)(-1/26))^3, ...

Simplify the progression:

243/26, -9/26, 81/676, -729/17576, ...

Conclusion

In this article, we explored how to determine an infinite geometric progression given the sum of the first three terms and the sum of the entire progression. We used the given information to solve for the common ratio and the first term, and then determined the progression itself. The progression is given by:

243/26, -9/26, 81/676, -729/17576, ...

This progression can be used to model a wide range of real-world phenomena, from population growth to financial investments.

References

[1] "Infinite Geometric Progression" by Math Open Reference
[2] "Geometric Progression" by Wolfram MathWorld

Introduction

In our previous article, we explored how to determine an infinite geometric progression given the sum of the first three terms and the sum of the entire progression. In this article, we will answer some of the most frequently asked questions about infinite geometric progressions.

Q: What is an infinite geometric progression?

A: An infinite geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Q: How do I calculate the sum of an infinite geometric progression?

A: The sum of an infinite geometric progression can be calculated using the formula:

S = a / (1 - r)

where 'a' is the first term and 'r' is the common ratio.

Q: What is the difference between a finite and infinite geometric progression?

A: A finite geometric progression is a sequence of numbers with a fixed number of terms, while an infinite geometric progression is a sequence of numbers with an infinite number of terms.

Q: Can I use an infinite geometric progression to model real-world phenomena?

A: Yes, infinite geometric progressions can be used to model a wide range of real-world phenomena, from population growth to financial investments.

Q: How do I determine the common ratio of an infinite geometric progression?

A: The common ratio of an infinite geometric progression can be determined using the formula:

r = (sum of the first three terms) / (sum of the entire progression)

Q: What is the significance of the common ratio in an infinite geometric progression?

A: The common ratio determines the rate at which the terms of the progression increase or decrease. If the common ratio is greater than 1, the terms of the progression will increase exponentially. If the common ratio is less than 1, the terms of the progression will decrease exponentially.

Q: Can I use an infinite geometric progression to model a population that is growing exponentially?

A: Yes, an infinite geometric progression can be used to model a population that is growing exponentially. The common ratio would represent the rate of growth of the population.

Q: How do I determine the first term of an infinite geometric progression?

A: The first term of an infinite geometric progression can be determined using the formula:

a = (sum of the entire progression) * (1 - r)

Q: What is the significance of the first term in an infinite geometric progression?

A: The first term of an infinite geometric progression represents the initial value of the sequence.

Q: Can I use an infinite geometric progression to model a financial investment that earns interest?

A: Yes, an infinite geometric progression can be used to model a financial investment that earns interest. The common ratio would represent the interest rate.

Conclusion

In this article, we answered some of the most frequently asked questions about infinite geometric progressions. We hope that this guide has provided you with a better understanding of these sequences and how they can be used to model real-world phenomena.

References

[1] "Infinite Geometric Progression" by Math Open Reference
[2] "Geometric Progression" by Wolfram MathWorld