Problem 23 - Contemporary Precalculus

Question

In Exercises \(23-30,\) show that the given sequence is geometric and find the common ratio. $$\left\\{\left(-\frac{1}{2}\right)^{n}\right\\}$$

Step-by-Step Solution

Answer

If so, what is the common ratio? Answer: Yes, the sequence is geometric, and the common ratio is \(-\frac{1}{2}\).

Step 1: Write down the sequence for the first few terms

Write the first few terms down by substituting n by 1, 2, 3, and 4 in the given formula: \((-\frac{1}{2})^{n}\). This will yield the following sequence: $$\left(-\frac{1}{2}\right),\ \left(-\frac{1}{2}\right)^{2},\ \left(-\frac{1}{2}\right)^{3},\ \left(-\frac{1}{2}\right)^{4},\ \dots$$

Step 2: Determine if there's a common ratio between the terms

To determine if there's a common ratio, we need to divide consecutive terms in the sequence. For example, dividing terms \(a_{2}\) by \(a_{1}\), \(a_{3}\) by \(a_{2}\), and so on. $$\frac{(-\frac{1}{2})^{2}}{(-\frac{1}{2})},\ \frac{(-\frac{1}{2})^{3}}{(-\frac{1}{2})^{2}},\ \frac{(-\frac{1}{2})^{4}}{(-\frac{1}{2})^{3}},\ \dots$$

Step 3: Simplify the ratios and compare them

Now, simplify the ratio of consecutive terms: $$\frac{(\frac{1}{4})}{(-\frac{1}{2})} = -\frac{1}{2},\ \frac{-\frac{1}{8}}{\frac{1}{4}} = -\frac{1}{2},\ \frac{\frac{1}{16}}{-\frac{1}{8}} = -\frac{1}{2},\ \dots$$ As we can see, each ratio is the same (-1/2) for the consecutive terms in the sequence. It means that the sequence is a geometric sequence.

Step 4: Write the common ratio

Since the ratio between consecutive terms is equal and found to be \(-1/2\), the sequence is geometric, and the common ratio is: $$r = -\frac{1}{2}$$

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a fundamental element that defines the sequence's behavior. This ratio remains constant between any two consecutive terms of the sequence. If you know the common ratio, you can predict any term in the sequence by multiplying the previous term by this fixed number. In the given problem, the sequence is defined by the formula \(-\frac{1}{2}\)^n\. When we calculate the common ratio, we divide one term by its predecessor. For example, dividing \(a_2\) by \(a_1\) gives us \(\frac{(-\frac{1}{2})^2}{(-\frac{1}{2})}\), which simplifies to -1/2.

• The consistency of the common ratio (-1/2) confirms that every term arises from multiplying the previous term by -1/2.
• This repetitive multiplication fully characterizes the sequence as geometric.

A key takeaway is that knowing the common ratio equips you to generate or identify any term within the sequence, which is invaluable for understanding patterns in mathematics.

Sequence Terms

Sequence terms are the individual elements that make up a sequence. In the context of a geometric sequence, these terms follow a distinct rule defined by the common ratio. Each term is obtained by the multiplication of the previous term with the common ratio. In our example, the sequence starts with the term \(-\frac{1}{2}\)\. Each subsequent term is generated by multiplying the preceding term by -1/2.

• The first term is \(-\frac{1}{2}\).
• The second term becomes \((-\frac{1}{2})^2 = \frac{1}{4}\).
• Continuing this process yields \((-\frac{1}{2})^3 = -\frac{1}{8}\), and so on.

Recognizing the pattern and calculating further terms becomes straightforward, thanks to the common ratio rule. Understanding sequence terms is essential in mathematics as it helps solve problems related to patterns and series.

Mathematics

Mathematics encompasses various fields, including the study of sequences. The exploration of geometric sequences is an essential part of algebra, which provides insight into how sequences behave over time. It teaches us to find patterns and understand the relationships between numbers. Learning about sequences helps sharpen analytical skills as well as mathematical intuition.

• Geometric sequences display exponential growth or decay due to the repeated multiplication by the common ratio.
• These sequences have real-world applications such as modeling population growth, interest calculations in finance, and understanding waves in physics.

As students become familiar with these concepts, they grasp fundamental mathematical principles, enabling them to apply this knowledge practically. By understanding the role of common ratios and sequence terms, they can analyze problems more efficiently, illustrating the interconnections within mathematics.