Problem 70 - Beginning and Intermediate Algebra

Question

Find the sum of the terms of the infinite geometric sequence, if possible. $$a_{1}=18, r=\frac{1}{3}$$

Step-by-Step Solution

Answer

The sum of the terms of the given infinite geometric sequence is 27.

Step 1: Identify if the sequence converges

Check if the sequence converges by confirming whether the absolute value of the common ratio is less than 1. Given: r = 1/3, As \(|r| = \frac{1}{3} < 1\), the sequence converges.

Step 2: Apply the formula

Using the formula for the sum S of an infinite geometric sequence which converges: \[S = \frac{a_1}{1 - r}\]

Step 3: Plug in the given values

Now, input the values we are given into the formula: \(a_1 = 18\) and \(r = \frac{1}{3}\) \[S = \frac{18}{1 - \frac{1}{3}}\]

Step 4: Simplify the expression

Simplify the denominator after substituting the values: \[S = \frac{18}{\frac{2}{3}}\]

Step 5: Calculate the sum

To get the sum S, multiply the numerator and denominator by 3: \[S = \frac{18 \times 3}{2}\] \[S = \frac{54}{2}\] \[S = 27\] So, the sum of the terms of the given infinite geometric sequence is 27.

Key Concepts

Convergence of Series

Determining whether a series converges is fundamental in understanding infinite series. For a geometric sequence, this depends on the common ratio \( r \). A geometric series will converge if the absolute value of this ratio is less than 1, meaning the terms in the series approach zero as they progress. If the sequence converges, we can find its sum. If it diverges, the terms increase indefinitely and a finite sum doesn't exist.

• The series converges when \(|r| < 1\).
• The sequence diverges when \(|r| \geq 1\).

In our example, where \( r = \frac{1}{3} \), the sequence converges since \(|r| = \frac{1}{3} < 1\). Knowing that the series converges is the first step before proceeding to find the sum of the series.

Sum of Infinite Series

Once we've established convergence, calculating the sum of an infinite geometric series becomes feasible. The sum represents what all the terms in the series would add up to, even though there are infinitely many of them. For convergent infinite geometric series, we use a specific formula to find this sum, ensuring the calculations are precise and understandable.
The formula to calculate the sum \( S \) is:\[S = \frac{a_1}{1-r}\]

• \(a_1\) is the first term in the series.
• \(r\) is the common ratio between successive terms.

In our example with \(a_1 = 18\) and \(r = \frac{1}{3}\), we plug these into the formula. This gives:\[S = \frac{18}{1 - \frac{1}{3}}\]Eventually simplifying to \( S = 27 \). Thus, the sum of this infinite series is 27.

Geometric Sequence Formula

The geometric sequence formula is a critical tool in managing series, especially when dealing with infinite terms. This formula helps us systematically understand and calculate elements within a sequence that increases or decreases by a constant ratio. A geometric sequence is defined by its first term and its common ratio, which plays a central role in determining the behavior and characteristics of the sequence including convergence.
The general form for any term in a geometric sequence is:\[a_n = a_1 \cdot r^{n-1}\]

• \(a_n\) represents the \(n\)-th term in the sequence.
• \(a_1\) is the first term.
• \(r\) is the common ratio.
• \(n\) is the term number.

The formula allows us to find any term in the sequence without listing all preceding terms. In our specific problem, understanding \(a_1 = 18\) and \(r = \frac{1}{3}\) enables not just computing specific terms but also the sum when the series converges.