Introduction to Geometric Progressions

A geometric progression (GP) 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. In this article, we will explore how to find the common ratio of an infinite GP and discuss some related concepts and methods.

Understanding the Problem

The problem posits that the first term of the GP is 1, and each term is equal to the sum of all the succeeding terms. This condition allows us to derive the common ratio of the GP.

Determining the Common Ratio

Let the common ratio of the GP be r. The terms of the GP can be expressed as follows:

First term: a 1 Second term: ar r Third term: ar^2 r^2 Fourth term: ar^3 r^3

The sum of the terms of an infinite GP is given by the formula:

S frac{a}{1 - r}

Given that each term is equal to the sum of the succeeding terms, we can set up the equation as follows:

1 r r^2 r^3 ldots

The sum of the series on the right can be expressed using the sum formula:

S frac{r}{1 - r}

Setting this equal to 1:

1 frac{r}{1 - r}

By cross-multiplying, we get:

1 - r r

This simplifies to:

1 2r implies r frac{1}{2}

Thus, the common ratio of the infinite GP is:

boxed{frac{1}{2}}

Alternative Methods for Finding the Common Ratio

Another method involves the use of the difference of successive terms or dividing the larger earlier term by the smaller successor. This method can also help in finding the common ratio r.

Using Division of Successive Terms

If we take the difference of any two successive terms and divide the larger earlier term by the smaller successor, the result will give us the common ratio r.

The relationship between terms in a GP can be expressed as:

t_n t_{n-1} t_{n-2} ldots S_infty - S_n

Where:

ar^{n-1} frac{a}{1-r} - aleft{1 - r^nright} / (1 - r) (r^{n-1}(1-r) 1 - frac{1}{r^n}) (r^{n-1} - r^n r^n) (r^{n-1} - r^n r^n) implies (r^{n-1} 2r^n) (r^{n-1}/r 2r^n) implies (1/r 2) (r 1/2)

Alternative Perspective: Golden Ratio

In some cases, the common ratio can be related to the solutions of the quadratic equation λ^2 - λ - 1 0, which are the roots of the equation. These roots are known as the Golden Ratio, which is approximately 1.618033988749895. The first term of the GP can be 1, and any term A_n A_{n-1} A_{n-2} represents the condition of the sum of succeeding terms.

Considering three terms of this GP as A/lambda, A, Alambda, we have:

A/lambda A (Alambda)

Where λ is the common ratio. Simplifying, we get:

1/lambda 1 λ implies lambda^2 lambda - 1 0

Solving the quadratic equation using the quadratic formula:

λ left[-1 pm sqrt{1 4}right]/2

This gives us two solutions:

λ frac{-1 sqrt{5}}{2} approx 0.618 λ frac{-1 - sqrt{5}}{2} approx -1.618 (which is not valid for a GP with positive terms)

Conclusion

The common ratio of the infinite GP can be derived using various methods, including the sum formula, the relationship between successive terms, and the solutions to a quadratic equation related to the Golden Ratio. This exploration provides a comprehensive understanding of how to find the common ratio and highlights the rich mathematical concepts underlying geometric progressions.