Determining the Common Ratio of an Infinite Geometric Series

In the study of mathematical sequences and series, one frequently encounters the concept of the infinite geometric series. This article will focus on a specific problem involving such a series: if the first term of an infinite geometric series is equal to twice the sum of all the terms that follow it, then we will determine the common ratio of the series. Let's explore the solution step by step.

Understanding the Problem and Mathematical Representation

Consider an infinite geometric series with the first term denoted as a and the common ratio as r. The sum of this infinite geometric series can be represented as:

S ( frac{a}{1 - r} ).

The problem states that the first term a is equal to twice the sum of all the terms that follow it. We can express the sum of the terms that follow the first term as:

S infinite - a ( frac{a}{1 - r} - a ).

Solving the Problem

Let's proceed step by step to solve the equation:

Set up the equation based on the given information:

a 2 ( frac{ar}{1 - r} )

Simplify the equation:

a ( frac{2ar}{1 - r} )

Assuming a ≠ 0, divide both sides by a to simplify further:

1 ( frac{2r}{1 - r} )

Cross-multiply to eliminate the fraction:

1 - r 2r

Rearrange the equation to solve for r:

1 3r

r ( frac{1}{3} )

Conclusion

In conclusion, the value of the common ratio r is:

( boxed{frac{1}{3}} )

Additional Insights

The solution presented here is a classic example that highlights the importance of understanding the properties of geometric series. Let's revisit this problem using a slightly different approach:

Given:

a 2Sinfinite - a

Let's rewrite Sinfinite S for simplicity:

a 2S - a

3a 2S

And given the sum of the series: S ( frac{2a}{1 - r} ), we can substitute and solve for r:

3a ( frac{2a}{1 - r} )

Following similar steps as before, we arrive at the same conclusion:

r ( frac{1}{3} )

Mathematical Representations and Notations

In the context of geometric series, notation is crucial. Let's rewrite the series using the standard notation:

Series: a, ar, ar2, ar3, …

The problem states:

a 2(ar ar2 ar3 …)

Let x the sum of all the terms that follow the first term:

x ar ar2 ar3 …

Multiply both sides by r:

rx ar2 ar3 …

Subtract the latter series from the former:

x - rx ar

Which simplifies to:

x ( frac{ar}{1-r} )

We know that the first term is twice the sum of the remaining terms:

a 2x 2 ( frac{ar}{1-r} )

Multiply both sides by ( frac{1-r}{a} )

1 - r 2r

Simplifying, we get:

r ( frac{1}{3} )

Further Reading and Resources

For more in-depth exploration of geometric series and their applications, readers are encouraged to refer to the following resources:

An Introduction to Geometric Series: An introductory article that covers the basics of geometric series and their properties. Application of Geometric Series in Real-World Scenarios: A discussion on how geometric series are applied to solve real-world problems in various fields, including finance and physics. Interactive Geometric Series Calculator: An online tool that helps you explore the behavior of geometric series with different common ratios and first terms.