Problem 27 - A Graphical Approach to Precalculus with Limits

Question

Use the formula for \(S_{n}\) to find the sum of the first five terms for each geometric sequence. Round the answers for Exercises 29 and 30 to the nearest hundredih. $$18,-9, \frac{9}{2},-\frac{9}{4}, \dots$$

Step-by-Step Solution

Answer

The sum of the first five terms is 12.38.

Step 1: Identify the First Term and Common Ratio

The geometric sequence is given as \(18, -9, \frac{9}{2}, -\frac{9}{4}, \dots\). The first term \(a\) is 18. The common ratio \(r\) can be found by dividing the second term by the first term: \(r = \frac{-9}{18} = -\frac{1}{2}\).

Step 2: Use the Formula for Sum of Geometric Sequence

The formula for the sum of the first \(n\) terms of a geometric sequence is \(S_n = a \frac{1-r^n}{1-r}\). Here, \(a = 18\), \(r = -\frac{1}{2}\), and \(n = 5\).

Step 3: Calculate the Common Ratio Raised to Power of Number of Terms

Compute \(r^n\) where \(n=5\): \((-\frac{1}{2})^5 = -\frac{1}{32}\).

Step 4: Substitute Values into the Formula

Substitute \(a = 18\), \(r = -\frac{1}{2}\), \(r^n = -\frac{1}{32}\) into the sum formula: \[S_5 = 18 \frac{1 - (-\frac{1}{32})}{1 - (-\frac{1}{2})}\] This simplifies to: \[S_5 = 18 \frac{1 + \frac{1}{32}}{1 + \frac{1}{2}} = 18 \frac{\frac{33}{32}}{\frac{3}{2}}\]

Step 5: Simplify the Expression

First calculate \(\frac{33}{32} \div \frac{3}{2}\): \[\frac{33}{32} \times \frac{2}{3} = \frac{66}{96} = \frac{11}{16}\]Now calculate the final result: \[S_5 = 18 \times \frac{11}{16} = \frac{198}{16} = 12.375\]

Step 6: Round the Result

Round the sum to the nearest hundredth: 12.38.

Key Concepts

Sum of Geometric Sequence

A geometric sequence is a series of numbers where each term is derived by multiplying the previous one by a constant value, known as the common ratio. The sum of the first few terms of a geometric sequence can be calculated using a specific formula. The formula is:\[S_n = a \frac{1-r^n}{1-r} \]where:

• \(S_n\) is the sum of the first \(n\) terms,
• \(a\) is the first term of the sequence,
• \(r\) is the common ratio,
• \(n\) is the number of terms you want to sum up.

This formula helps us calculate the sum of the sequence without having to add each term individually. By substituting values into the formula, you can find the sum for any number of terms. In our example, the first five terms of the sequence sum up to 12.38.

Common Ratio

The common ratio in a geometric sequence is the factor by which each term is multiplied to get the next term. It is a fundamental characteristic of a geometric sequence. Finding the common ratio \(r\) is crucial for solving and understanding the sequence.To determine the common ratio, divide one term in the sequence by the preceding term. For example, if the sequence progresses as 18, -9, \(\frac{9}{2}\), each term is derived by multiplying the previous term by the common ratio \(-\frac{1}{2}\). This is calculated as:\[r = \frac{-9}{18} = -\frac{1}{2}\]Knowing the common ratio allows you to predict subsequent terms and leverage the sum formula effectively.

First Term

The first term of a geometric sequence is denoted by \(a\), and it is the starting point from which all other terms are derived using the common ratio. Understanding the role of the first term is important because it serves as the basis for calculating further terms and applying the sum formula.In the example provided, the first term is 18. Starting from this initial value, subsequent terms are calculated by repeatedly applying the common ratio. As such, the first term significantly influences the values of the subsequent terms and the sum of the sequence. Always identify and confirm the first term accurately before using it in any calculations involving the sequence or its sum.