Problem 16 - Algebra 2

Question

Find the sum of each infinite geometric series, if it exists. \(a_{1}=18, r=0.6\)

Step-by-Step Solution

Answer

The sum of the series is 45.

Step 1: Understand the Infinite Geometric Series Formula

The sum of an infinite geometric series can be found using the formula: \[S = \frac{a_1}{1 - r}\]where \(a_1\) is the first term and \(r\) is the common ratio. This formula is valid only if \(|r| < 1\). In this problem, \(a_1 = 18\) and \(r = 0.6\), which satisfies \(|r| < 1\).

Step 2: Apply the Infinite Geometric Series Formula

Substitute the given values of \(a_1\) and \(r\) into the formula: \[S = \frac{18}{1 - 0.6}\]Calculate the denominator: \(1 - 0.6 = 0.4\).

Step 3: Calculate the Sum of the Series

Substitute the calculated denominator back into the formula to find the sum:\[S = \frac{18}{0.4}\]Divide 18 by 0.4 to get the sum:\[S = 45\]

Key Concepts

Geometric Series Formula

An infinite geometric series is a sequence where each term is created by multiplying the previous term by a constant called the common ratio, denoted as \(r\). When we calculate the sum of such a series, we use a specific formula. The geometric series formula to find the sum \(S\) of an infinite series is:

• \(S = \frac{a_1}{1 - r}\)

where \(a_1\) is the first term of the series. It's important to note that this formula is applicable only when the absolute value of the common ratio is less than one (\(|r| < 1\)). When \(|r|\) is less than one, the series converges to a finite value, allowing us to calculate its sum. In our problem, the first term \(a_1 = 18\) and the common ratio \(r = 0.6\), and since \(0.6 < 1\), we can confidently apply the formula.

Sum of Series

Calculating the sum of an infinite geometric series involves a few clear steps. Once you confirm that the series can indeed converge by checking \(|r| < 1\), you can plug the values into the geometric series formula. For our given series:

• First term \(a_1 = 18\)
• Common ratio \(r = 0.6\)

Substituting these into the formula gives:

• \(S = \frac{18}{1 - 0.6}\)
• \(S = \frac{18}{0.4}\)

The first step is to calculate the denominator, which is \(1 - r\). Here, it is \(1 - 0.6 = 0.4\). Hence, dividing the first term 18 by 0.4 results in the sum of the series:

• \(S = 45\)

This tells us that the series converges to and sums up to the value 45.

Common Ratio

The common ratio \(r\) is a crucial element in any geometric series. It determines how each term in the series relates to its preceding term. Calculated by dividing any term by its previous term, the value of \(r\) defines the progression of the series. In our example, \(r = 0.6\). This ratio indicates that every term is 60% of the term before it.
For an infinite geometric series to have a sum, \(r\) must satisfy the condition \(|r| < 1\). If \(|r| \geq 1\), the series either diverges or repeats indefinitely, making the sum impossible to compute. Hence, the smaller the absolute value of \(r\), the more quickly the terms approach zero, allowing the series to converge.

Convergence of Series

Understanding the convergence of an infinite geometric series is key to determining whether we can find its sum. Convergence occurs when the terms of the series approach zero as you proceed to infinity. For a geometric series, this happens when the common ratio \(r\) is such that \(|r| < 1\).
When a series converges, it settles to a particular finite sum, which we can calculate using the geometric series formula. In the problem at hand, with \(r = 0.6\), we know it converges because 0.6 is less than 1. The series stabilizes and sums up to 45, ensuring it doesn't grow indefinitely. Conversely, if \(r\) had been greater than or equal to 1, the series would diverge, meaning we couldn't assign it a finite sum, as the terms wouldn't diminish to zero.