Problem 26 - Calculus Early Transcendentals: Pearson New International Edition

Question

Prove the following formula for a geometric sum: $$ \sum_{k=0}^{n} a r^{k}=a+a r+a r^{2}+\cdots+a r^{n}=\frac{a-a r^{n+1}}{1-r}(r \neq 1) $$

Step-by-Step Solution

Answer

The geometric sum formula is valid and is \( S = \frac{a - ar^{n+1}}{1-r} \) when \( r \neq 1 \).

Step 1: Define the Geometric Series

Consider the finite geometric series given as \( S = a + ar + ar^2 + \cdots + ar^n \). Our goal is to find an expression for this sum.

Step 2: Multiply the Series by the Common Ratio

Multiply the entire series \( S \) by the common ratio \( r \):\[ rS = ar + ar^2 + ar^3 + \cdots + ar^{n+1} \].

Step 3: Subtract the Two Equations

Subtract the equation for \( rS \) from the equation for \( S \):\[ S - rS = (a + ar + ar^2 + \cdots + ar^n) - (ar + ar^2 + ar^3 + \cdots + ar^{n+1}) = a - ar^{n+1} \].

Step 4: Factor and Simplify

After subtraction, the left side becomes \( S(1-r) \). Thus, \\[ S(1-r) = a - ar^{n+1} \], hence, \\[ S = \frac{a - ar^{n+1}}{1-r} \].

Step 5: Ensure the Validity Condition

The formula is valid only when \( r eq 1 \), as dividing by zero is undefined.

Key Concepts

Geometric Sum Formula

The geometric sum formula provides a way to calculate the sum of a series of terms where each term is a constant multiple of the previous one. This type of series, known as a geometric series, can be expressed as \( S = a + ar + ar^2 + \cdots + ar^n \). Here, \( a \) represents the first term and \( r \) is the common ratio. The formula for calculating this sum is:\[ S = \frac{a - ar^{n+1}}{1-r} \]This formula is particularly useful because it simplifies the process of summing terms exponentially growing or decreasing at a constant rate. To make sure we use this equation correctly, note that it applies only when \( r eq 1 \), because if \( r = 1 \), the series becomes a simple arithmetic sum.

Proof by Subtraction

To derive the geometric sum formula, a method called proof by subtraction is used. This involves manipulating the series algebraically to isolate the desired expression. Here’s how it works:

• Start with the original sum \( S = a + ar + ar^2 + \cdots + ar^n \).
• Multiply each term in the series by the common ratio \( r \), giving: \( rS = ar + ar^2 + ar^3 + \cdots + ar^{n+1} \).
• Subtract this new equation from the original sum: \( S - rS = a - ar^{n+1} \).

This subtraction effectively cancels out most of the terms, leaving only the first term of \( S \) and the last term of \( -rS \), simplifying the sum dramatically. This approach clearly showcases the elegance and utility of mathematical manipulation.

Common Ratio in Series

The common ratio, denoted as \( r \), is a crucial element of geometric series. It is the constant factor between consecutive terms of the series. For the series \( a, ar, ar^2, \cdots \), the common ratio is \( r \).

• If \( |r| < 1 \), the terms get progressively smaller, and the series converges.
• If \( |r| > 1 \), the terms increase exponentially.
• The series is undefined for \( r = 1 \) under the geometric series formula because it simplifies instead to an arithmetic progression.

Understanding the common ratio helps predict how the terms of the series will behave, which is essential for applying the geometric sum formula accurately.

Finite Geometric Series

A finite geometric series is one that contains a limited number of terms, unlike an infinite geometric series which goes on indefinitely. In this context, the series is given as:\( S = a + ar + ar^2 + \cdots + ar^n \)In a finite geometric series, the number of terms is \( n+1 \). The simplicity of the geometric sum formula makes it easy to compute the sum of all these terms without having to add each individually.Some key points about a finite geometric series include:

• The formula for the sum can only be used when \( r eq 1 \).
• In practical applications, finite series are common in disciplines like economics and physics where exponential growth or decay sectors are analyzed.

Thus, the concept of a finite geometric series and its formula simplify real-world applications, providing a concise calculation method for complex problems.