Problem 55 - College Algebra

Question

The sum of an infin te geometric series is five times the value of the fi st term. What is the common ratio of the series?

Step-by-Step Solution

Answer

The common ratio is \( \frac{4}{5} \).

Step 1: Understand the Formula for Infinite Geometric Series

The formula for the sum of an infinite geometric series is \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio, which is between -1 and 1.

Step 2: Set Up the Equation

According to the problem, the sum of the series, \( S \), is five times the first term, \( a \). Therefore, the equation is \( \frac{a}{1 - r} = 5a \).

Step 3: Simplify the Equation

Divide both sides by \( a \) (assuming \( a eq 0 \)) to simplify: \( \frac{1}{1 - r} = 5 \).

Step 4: Solve for the Common Ratio

Rearrange the equation to find \( r \). Multiply both sides by \( 1 - r \) to get \( 1 = 5(1 - r) \).

Step 5: Expand and Isolate the Common Ratio

Expand the right side: \( 1 = 5 - 5r \). Move \( 5 \) to the left side: \( 1 - 5 = -5r \).

Step 6: Solve for \( r \)

Simplifying the left side gives \( -4 = -5r \). Divide both sides by \( -5 \) to solve for \( r \), so \( r = \frac{4}{5} \).

Key Concepts

Understanding Common Ratio

In an infinite geometric series, the common ratio is a fundamental element that determines how the series progresses. This ratio, often denoted by \( r \), is the factor by which we multiply consecutive terms to generate the next term in the series. For example, if each term is being multiplied by \( \frac{1}{2} \) to get to the next, the common ratio is \( \frac{1}{2} \). Understanding the common ratio is crucial because it influences whether a series will converge (have a finite sum) or diverge (sum to infinity). Typically, the common ratio needs to fall between \(-1\) and \(1\) for the series to converge, meaning the terms shrink to zero as we progress along the series.
By determining the common ratio, one can ascertain the behavior of the series over time, and predict other properties such as its sum for the case of a convergent series.

Exploring the Sum of Series Formula

The sum of an infinite geometric series with a first term \( a \) and common ratio \( r \) (where \( |r| < 1 \)) is given by the formula \( S = \frac{a}{1-r} \). This formula allows us to calculate a finite sum of the infinite series. The fascinating part of this is that although an infinite series has limitless terms, if the common ratio is between \(-1\) and \(1\), the series can sum to a finite number!
Therefore, the sum of the series depends greatly on the values of \( a \) and \( r \). If either is not consistent with the requirements of the formula, the series won’t converge, meaning no finite sum can be determined.

Setting Up Algebraic Equations

Algebraic equations give us a structured way to express mathematical relationships and solve problems. When dealing with the sum of an infinite geometric series, often you'll set up an algebraic equation that correlates given conditions with the sum formula. For instance, when the problem states that the sum is five times the first term, you translate this into an equation like \( \frac{a}{1-r} = 5a \).
This equation essentially describes a balance between the series formula and the given condition. The ability to craft such equations from word problems is a key skill in algebra, as it allows you to solve for unknown variables efficiently.

Solving Equations to Find Unknowns

In solving equations, our goal is often to isolate the unknown variable. Here, we need to solve for the common ratio \( r \). Let's revisit the equation \( \frac{a}{1-r} = 5a \). By simplifying this equation (dividing by \( a \), assuming \( a eq 0 \)), we get \( \frac{1}{1-r} = 5 \).
Solving this involves:

• Multiplying both sides by \( 1-r \) to clear the fraction: \( 1 = 5(1-r) \).
• Expanding the equation: \( 1 = 5 - 5r \).
• Rearranging to isolate \( r \): Subtract \( 5 \) from both sides, yielding \( 1 - 5 = -5r \).
• Finally, solving for \( r \) by dividing by \(-5\), resulting in \( r = \frac{4}{5} \).

This step-by-step approach ensures you solve for \( r \) clearly and accurately, applying algebraic methods to reach a solution.