Problem 55 - College Algebra with Corequisite Support

Question

The sum of an infinite geometric series is five times the value of the first 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 Sum of 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 of the series.

Step 2: Set Up the Equation

According to the problem, the sum of the series is five times the first term. Thus, we have \( S = 5a \). Using the formula from step 1, we replace \( S \) with \( \frac{a}{1 - r} \) and set it equal to \( 5a \). So, \( \frac{a}{1 - r} = 5a \).

Step 3: Simplify the Equation

To eliminate \( a \) from the equation, divide both sides by \( a \) (assuming \( a eq 0 \)). This results in \( \frac{1}{1 - r} = 5 \).

Step 4: Solve for the Common Ratio

To find \( r \), solve the equation \( \frac{1}{1 - r} = 5 \). Take the reciprocal of both sides to get \( 1 - r = \frac{1}{5} \). Then, solve for \( r \) by rearranging the equation: \( r = 1 - \frac{1}{5} = \frac{4}{5} \).

Key Concepts

Infinite Series

An infinite series is a sum of terms that continues indefinitely. It's like taking an endless sequence of numbers and adding them together. But how can we add up infinite numbers? In the world of mathematics, we often deal with this by looking at the concept of convergence. When an infinite series converges, it approaches a specific value as the number of terms increases. A common type of infinite series we encounter is the geometric series, where each term is a fixed multiple of the previous term. This is controlled by the common ratio, a crucial number that determines the series' behavior. In geometric series specifically, if the absolute value of the common ratio is less than 1, the series converges, meaning it adds up to some finite number.

Sum of Series

Finding the sum of an infinite geometric series involves using a clever formula. The sum, denoted by \( S \), is given by \( S = \frac{a}{1 - r} \). Here, \( a \) is the first term of the series, and \( r \) is the common ratio. This formula is useful because it transforms the problem of summing an infinite number of terms into a simple algebraic calculation. For the formula to be valid, we need to ensure that the common ratio \( r \) satisfies \(|r| < 1\), which ensures convergence. If the series does converge, the sum \( S \) can give us a precise answer, which might seem surprising given the infinite nature of the series.

Common Ratio

The common ratio \( r \) in a geometric series is the factor by which each term is multiplied to get the next term. For example, in the series 2, 4, 8, the common ratio is 2, because each term is double the previous one. The value of the common ratio determines whether an infinite geometric series converges or diverges. If \(|r| < 1\), the series converges to a finite sum. If \(|r| \geq 1\), the series diverges, meaning it does not sum to a finite amount. In our exercise, finding \( r \) involved solving the equation \( \frac{a}{1 - r} = 5a \) after setting the sum equal to five times the first term. Simplifying gives \( r = \frac{4}{5} \), indicating that with \( r \) less than 1, the series converges to the given sum.