The proof of the geometric series formula is a fundamental concept in mathematics, particularly in calculus and algebra. It demonstrates the relationship between the sum of an infinite geometric series and its first term and common ratio. This formula has wide-ranging applications in various fields, including finance, physics, and computer science. Understanding the proof not only provides insight into the formula itself but also enhances the comprehension of infinite series and their properties. This article will delve into the proof of the geometric series formula, exploring its derivation and providing clear explanations to facilitate understanding.
Understanding the Geometric Series
Before delving into the proof of the geometric series formula, it's crucial to define what a geometric series is. A geometric series is a sequence of numbers where each term is obtained by multiplying the previous term by a constant factor called the common ratio. The general form of a geometric series is:
a + ar + ar^2 + ar^3 + ...
where:
- a is the first term
- r is the common ratio
The proof of the geometric series formula focuses on finding the sum of this infinite series.
Proof by Induction
One method to prove the geometric series formula is using mathematical induction. This approach involves proving the formula for the base case (n = 1) and then proving that if it holds for any arbitrary value of n, it also holds for n + 1.
Base Case (n = 1)
For n = 1, the geometric series becomes:
a
The sum of this single term is simply 'a', and the formula for the sum of a geometric series, S = a(1 - r^n) / (1 - r), also yields 'a' when n = 1:
S = a(1 - r^1) / (1 - r) = a(1 - r) / (1 - r) = a
Therefore, the formula holds for the base case.
Inductive Step
Assume that the formula holds for an arbitrary value of n, meaning the sum of the first n terms is:
S_n = a(1 - r^n) / (1 - r)
We need to prove that the formula also holds for n + 1, meaning the sum of the first n + 1 terms is:
S_{n+1} = a(1 - r^(n+1)) / (1 - r)
The sum of the first n + 1 terms can be expressed as the sum of the first n terms plus the (n + 1)th term:
S_{n+1} = S_n + ar^(n+1)
Substituting the formula for S_n, we get:
S_{n+1} = a(1 - r^n) / (1 - r) + ar^(n+1)
Simplifying the expression by finding a common denominator:
S_{n+1} = [a(1 - r^n) + ar^(n+1)(1 - r)] / (1 - r)
Expanding the terms:
S_{n+1} = [a - ar^n + ar^(n+1) - ar^(n+2)] / (1 - r)
Rearranging the terms:
S_{n+1} = [a - ar^(n+2)] / (1 - r)
Factoring out 'a' and simplifying:
S_{n+1} = a(1 - r^(n+1)) / (1 - r)
This is the formula for the sum of the first n + 1 terms, which confirms that the formula holds for n + 1 if it holds for n.
Proof by Direct Calculation
Another method to prove the geometric series formula involves directly calculating the sum of the series. This method involves multiplying the series by the common ratio and subtracting the original series.
Let's denote the sum of the geometric series by 'S':
S = a + ar + ar^2 + ar^3 + ...
Multiplying both sides by the common ratio 'r':
rS = ar + ar^2 + ar^3 + ar^4 + ...
Subtracting the second equation from the first:
S - rS = a
Simplifying the equation:
S(1 - r) = a
Solving for S:
S = a / (1 - r)
This formula represents the sum of an infinite geometric series where the absolute value of the common ratio is less than 1 (|r| < 1). If |r| ≥ 1, the series diverges and does not have a finite sum.
Conclusion
The proof of the geometric series formula demonstrates the powerful relationship between the sum of an infinite geometric series and its first term and common ratio. Both methods, proof by induction and proof by direct calculation, provide insightful approaches to understanding the formula's derivation. This formula is a cornerstone of various mathematical and scientific concepts, and its understanding is essential for tackling problems related to infinite series, growth, and decay phenomena.
