Derive and Use the Formula for the Sum of a Finite Geometric Series

Introduction

Key Concepts

Understanding Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant ratio. This ratio is a key characteristic that defines the nature of the series.

Mathematically, a geometric series can be expressed as: $$ a, ar, ar^2, ar^3, \ldots, ar^{n-1} $$ where:

• a is the first term.
• r is the common ratio.
• n is the number of terms.

Sum of a Finite Geometric Series

The sum of a finite geometric series is the total obtained by adding all the terms of the series up to a specified number of terms. Deriving the formula for this sum is essential for solving various algebraic problems.

Consider the series: $$ S_n = a + ar + ar^2 + \ldots + ar^{n-1} $$ To derive the formula for \( S_n \), follow these steps:

1. Multiply the entire series by the common ratio \( r \): $$ rS_n = ar + ar^2 + ar^3 + \ldots + ar^n $$
2. Subtract the second equation from the first: $$ S_n - rS_n = a - ar^n $$
3. Simplify the equation: $$ S_n(1 - r) = a(1 - r^n) $$
4. Solve for \( S_n \): $$ S_n = \frac{a(1 - r^n)}{1 - r} \quad \text{for} \quad r \neq 1 $$

This formula allows us to calculate the sum of the first \( n \) terms of a geometric series efficiently.

Applying the Formula

Once the formula is established, it can be applied to various problems. For example, to find the sum of the first 5 terms of a geometric series with \( a = 3 \) and \( r = 2 \): $$ S_5 = \frac{3(1 - 2^5)}{1 - 2} = \frac{3(1 - 32)}{-1} = \frac{3(-31)}{-1} = 93 $$

Properties of Geometric Series

• If \( |r| < 1 \), the series converges as \( n \) approaches infinity.
• If \( |r| > 1 \), the series diverges.
• The formula for the sum changes when extending to an infinite series.

Real-World Applications

Geometric series have numerous applications, including calculating compound interest, population growth models, and analyzing algorithms in computer science. Understanding the sum of such series allows for the modeling and prediction of exponential growth or decay processes.

Advanced Concepts

Theoretical Derivations and Proofs

Beyond the basic formula, it's important to understand the underlying principles that make the formula valid. The derivation relies on the properties of geometric sequences and the principles of algebraic manipulation.

Consider the infinite geometric series where \( |r| < 1 \): $$ S = a + ar + ar^2 + ar^3 + \ldots $$ By multiplying both sides by \( r \): $$ rS = ar + ar^2 + ar^3 + \ldots $$ Subtracting the second equation from the first: $$ S - rS = a \\ S(1 - r) = a \\ S = \frac{a}{1 - r} $$ This shows the sum converges to \( \frac{a}{1 - r} \) when \( |r| < 1 \).

Complex Problem-Solving

Let's tackle a more complex problem involving nested geometric series:

Problem: Find the sum of the series: $$ S = 3 + 6 + 12 + 24 + \ldots + 768 $$ Solution:

• Identify \( a = 3 \) and \( r = 2 \).
• Find \( n \) such that \( ar^{n-1} = 768 \): $$ 3 \times 2^{n-1} = 768 \\ 2^{n-1} = \frac{768}{3} = 256 \\ 2^{n-1} = 2^8 \\ n - 1 = 8 \\ n = 9 $$
• Apply the sum formula: $$ S_9 = \frac{3(1 - 2^9)}{1 - 2} = \frac{3(1 - 512)}{-1} = \frac{3(-511)}{-1} = 1533 $$

Interdisciplinary Connections

Geometric series intersect with various disciplines:

• Finance: Calculating compound interest involves geometric series.
• Computer Science: Analyzing time complexities in algorithms, such as binary search.
• Physics: Modeling exponential decay in radioactive substances.

Extending to Infinite Series

While the focus is on finite series, extending to infinite series provides deeper insights:

• If \( |r| < 1 \), the infinite series sum is \( \frac{a}{1 - r} \).
• This concept is pivotal in calculus, particularly in studying convergence and limits.

Applications in Technology

In technology, geometric series underpin algorithms that divide problems into exponentially smaller subproblems, such as in divide and conquer strategies. Understanding these series aids in optimizing performance and resource allocation.

Proof by Mathematical Induction

To validate the sum formula, mathematical induction can be employed:

• Base Case: For \( n = 1 \), $$ S_1 = a \\ \frac{a(1 - r^1)}{1 - r} = \frac{a(1 - r)}{1 - r} = a $$ The base case holds.
• Inductive Step: Assume the formula holds for \( n = k \): $$ S_k = \frac{a(1 - r^k)}{1 - r} $$ For \( n = k + 1 \): $$ S_{k+1} = S_k + ar^k = \frac{a(1 - r^k)}{1 - r} + ar^k = \frac{a(1 - r^{k+1})}{1 - r} $$ Hence, the formula holds for \( n = k + 1 \).

Non-integer Ratios and Alternative Applications

Geometric series are not limited to integer ratios. For instance, financial models often involve fractional growth rates:

• Compound interest with \( r = 1.05 \) for 5% growth.
• Depreciation models with \( r = 0.9 \) representing a 10% decrease.

Comparison Table

Summary and Key Takeaways