Problem 24 - A Graphical Approach to College Algebra

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 25 and 26 to the nearest hundredth. $$12,-4, \frac{4}{3},-\frac{4}{9}, \dots$$

Step-by-Step Solution

Answer

The sum of the first five terms is approximately 9.04.

Step 1: Identify Geometric Sequence Elements

A geometric sequence is defined by the first term (\(a_1\)) and the common ratio (\(r\)). For the sequence given (\(12, -4, \frac{4}{3}, -\frac{4}{9}, \ldots\)), the first term \(a_1 = 12\).

Step 2: Determine Common Ratio

The common ratio \(r\) is found by dividing the second term by the first term: \(-4 \div 12 = \frac{-1}{3}\).

Step 3: Apply Sum Formula for Sn

The sum of the first \(n\) terms of a geometric sequence is given by the formula: \[S_n = a_1 \frac{1 - r^n}{1 - r}\] Substitute \(n = 5\), \(a_1 = 12\), and \(r = -\frac{1}{3}\).

Step 4: Calculate \(r^n\)

Find \((-\frac{1}{3})^5\). Since \((-\frac{1}{3})^5 = -\frac{1}{243}\), the value is needed for the next calculation.

Step 5: Plug Values into Formula

Substitute the values into the sum formula: \[S_5 = 12 \frac{1 - (-\frac{1}{243})}{1 - (-\frac{1}{3})}\].

Step 6: Simplify the Expression

Calculate the expression: \[1 - (-\frac{1}{243}) = \frac{244}{243}\] and \[1 + \frac{1}{3} = \frac{4}{3}\]. Substitute back: \[S_5 = 12 \frac{\frac{244}{243}}{\frac{4}{3}}\].

Step 7: Final Calculation

Multiply and simplify the expression: \[S_5 = 12 \times \frac{244}{243} \times \frac{3}{4} = 9.04\] (rounded to the nearest hundredth).

Key Concepts

Sum Formula for Geometric Sequence

When calculating the sum of the first few terms in a geometric sequence, we use a special formula. The sum formula for the first \(n\) terms of a geometric sequence is expressed as:\[S_n = a_1 \frac{1 - r^n}{1 - r}\]where \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms you want to sum. This formula derives from the pattern of multiplying each term by the common ratio. It's helpful because it allows you to calculate large sums quickly without manually adding each term.

• \(S_n\): sum of the first \(n\) terms
• \(a_1\): first term of the sequence
• \(r\): common ratio
• \(n\): number of terms in the sequence

To use this formula, you need to determine the values of both \(a_1\) and \(r\), which are essential components of any geometric sequence. Let's see how we can use this formula effectively for accurate calculations.

Common Ratio

In any geometric sequence, each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the "common ratio". This ratio, denoted as \(r\), is a key element of the sequence's structure. To find the common ratio, simply divide any term in the sequence by the term that comes just before it.

• Formula to calculate \(r\): \(r = \frac{a_2}{a_1}\)
• \(a_2\): second term
• \(a_1\): first term

For example, in the geometric sequence \(12, -4, \frac{4}{3}, -\frac{4}{9}, \ldots\), the common ratio is found by dividing the second term by the first: \(-4 \div 12 = \frac{-1}{3}\). The common ratio can be a fraction, a decimal, or even a negative number, which results in alternating signs in the sequence. Understanding the common ratio helps in predicting future terms and calculating sums efficiently.

Geometric Series

A geometric series is the sum of the terms of a geometric sequence. When dealing with a finite series, we can find the sum using the sum formula explained earlier. A key characteristic of a geometric series is that each term is a constant multiple of the preceding term; this constant is known as the common ratio. Consider the geometric series formed by the sequence \(12, -4, \frac{4}{3}, -\frac{4}{9}, \ldots\). Here, you're adding these numbers together to find the sum of the first few terms, which is a common task in math courses.

• Finite geometric series have a set number of terms.
• The sum of the series is what distinguishes it from just listing the sequence.

Using the sum formula, you can quickly calculate the sum, which in our example for the first five terms results in \(9.04\). This efficiency is what makes geometric series an important concept in both mathematics and practical applications, like finance and science.