Problem 23 - 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. $$18,-9, \frac{9}{2},-\frac{9}{4}, \dots$$

Step-by-Step Solution

Answer

The sum of the first five terms is 12.38.

Step 1: Identify the Sequence

We identify that the given sequence is geometric with the first term \(a = 18\) and the common ratio \(r = \frac{-9}{18} = -\frac{1}{2}\).

Step 2: Determine the Formula for the Sum of a Geometric Sequence

The formula for the sum of the first \(n\) terms of a geometric sequence is given by \(S_n = a \frac{1-r^n}{1-r}\), where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.

Step 3: Substitute the Known Values

Using \(a = 18\), \(r = -\frac{1}{2}\), and \(n = 5\), we substitute these into the sum formula: \(S_5 = 18 \frac{1-(-\frac{1}{2})^5}{1-(-\frac{1}{2})}\).

Step 4: Calculate the Sum for 5 Terms

First, calculate \((-\frac{1}{2})^5 = -\frac{1}{32}\). Substitute in to get: \(S_5 = 18 \frac{1 - (-\frac{1}{32})}{1 + \frac{1}{2}}\).

Step 5: Simplify the Expression

Simplify the expression: \(S_5 = 18 \frac{1 + \frac{1}{32}}{1.5}\). This becomes \(S_5 = 18 \frac{\frac{33}{32}}{\frac{3}{2}}\).

Step 6: Perform Final Calculations

Simplify further: \(S_5 = 18 \times \frac{33}{32} \times \frac{2}{3}\). This simplifies to \(S_5 = 18 \times \frac{66}{96}\), and finally \(S_5 = 12.375\).

Step 7: Round the Answer

Round 12.375 to the nearest hundredth to get \(S_5 = 12.38\).

Key Concepts

Sum of a Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The formula to find the sum of the first \( n \) terms of a geometric sequence is incredibly useful. The formula is:

• \( S_n = a \frac{1-r^n}{1-r} \)

Here:

• \( S_n \) 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.

The formula works best when \( r eq 1 \) because if \( r = 1 \), each term of the sequence is the same, and you would simply multiply the first term by \( n \). This formula simplifies the process of adding terms in a geometric sequence, saving you time and effort compared to calculating each term individually and adding them up. It provides a tidy package to solve such problems efficiently.
To apply the formula correctly, identify each variable correctly, and substitute the values accurately.

Common Ratio

In any geometric sequence, the common ratio is a crucial component. It's the factor you multiply each term by to get to the next term.
Let's say you have a sequence where the first term is 18 and the second is \(-9\). To find the common ratio, divide the second term by the first:

• \(-9 / 18 = -1/2\)

This division gives you the common ratio \( r \). It is important because it defines the behavior of the sequence, whether it grows by becoming larger or smaller, or even oscillates.
In our example, because \( r = -1/2 \), a negative ratio, the sequence's terms will alternate in sign. Understanding the common ratio helps in predicting how the sequence progresses over time. Being able to identify and compute it is vital in solving geometric sequence problems.

First Term

The first term in a geometric sequence, denoted as \( a \), is your starting point.
It is the initial term from which all subsequent terms of the sequence are derived using the common ratio. Having a clear understanding of the first term is fundamental, as it anchors the sequence.
For example, in our sequence, the first term \( a \) is noted explicitly as 18. Each subsequent term is produced by multiplying this first term successively by the common ratio.
This first term is pivotal because:

• It is input into the sum formula to find the sum of terms,
• It serves as the reference point for finding other terms.

Grasping this concept allows you to build and manipulate the sequence as needed.

Rounding to the Nearest Hundredth

Rounding is a significant mathematical skill, especially when dealing with complex calculations that result in lengthy decimals. When you're asked to round to the nearest hundredth, you are focusing on two digits after the decimal point.
You must look at the third digit to determine how the second digit will change:

• If the third digit is 5 or more, increase the second digit by 1.
• If it’s less than 5, keep the second digit the same.

For example, if you calculate a sum and get 12.375, you would look at the third decimal place (5 in this case). Since it's 5, you round up the previous number (7) to 8, resulting in 12.38. This step is essential for providing a precise and clear answer in your work, particularly in mathematics where accuracy of the result is crucial for subsequent calculations.