Problem 18 - Precalculus: Mathematics for Calculus

Question

The first four terms of a sequence are given. Determine whether these terms can be the terms of a geometric sequence. If the sequence is geometric, find the common ratio. $$27,-9,3,-1, \dots$$

Step-by-Step Solution

Answer

The sequence is geometric with a common ratio of \(-\frac{1}{3}\).

Step 1: Understand the Definition of a Geometric Sequence

A geometric sequence is a sequence of numbers where the ratio between consecutive terms is constant. This constant ratio is called the common ratio.

Step 2: Identify Given Terms

The given terms of the sequence are: 27, -9, 3, and -1. We need to find if these terms consistently have the same ratio when divided by their preceding terms.

Step 3: Calculate Ratio Between First and Second Term

To find the ratio \( r_1 \) between the first two terms, divide the second term by the first term: \( r_1 = \frac{-9}{27} = -\frac{1}{3} \).

Step 4: Calculate Ratio Between Second and Third Term

Find the ratio \( r_2 \) between the second and third terms: \( r_2 = \frac{3}{-9} = -\frac{1}{3} \).

Step 5: Calculate Ratio Between Third and Fourth Term

Find the ratio \( r_3 \) between the third and fourth terms: \( r_3 = \frac{-1}{3} = -\frac{1}{3} \).

Step 6: Verify Consistency of the Common Ratio

Since \( r_1 = r_2 = r_3 = -\frac{1}{3} \), the sequence has a consistent common ratio. This means the sequence is geometric.

Step 7: Conclusion

The sequence is a geometric sequence with a common ratio of \(-\frac{1}{3}\).

Key Concepts

Understanding the Common Ratio

In the world of geometric sequences, the **common ratio** plays a pivotal role. It's the factor by which we multiply one term to get to the next. Imagine you have a sequence, and each term is obtained by multiplying the previous one by this magical number, the common ratio.

If the common ratio remains constant throughout the entire sequence, then you've got yourself a geometric sequence!

• The common ratio is represented by the symbol \( r \).
• To find \( r \), divide any term by the preceding term.
• In a geometric sequence, all pairs of consecutive terms will share this ratio.

For example, given the sequence terms 27, -9, 3, and -1, you'll divide each by its predecessor:

• \( r_1 = \frac{-9}{27} = -\frac{1}{3} \)
• \( r_2 = \frac{3}{-9} = -\frac{1}{3} \)
• \( r_3 = \frac{-1}{3} = -\frac{1}{3} \)

Since these calculations yield a consistent ratio of \(-\frac{1}{3}\), this sequence exhibits a common ratio, proving it to be geometric.

Exploring Sequence Terms

The **sequence terms** are the individual values or numbers in a sequence. Every term in a sequence is connected and, in the case of a geometric sequence, is generated by multiplying the preceding term by the common ratio. This inherent pattern is what distinguishes geometric sequences from other types.

In our example, the first four sequence terms were 27, -9, 3, and -1. Let’s look a bit deeper:

• The first term is 27. This is typically where you start.
• The second term, -9, is obtained by multiplying 27 with the common ratio \(-\frac{1}{3}\).
• The third term is 3, a product of -9 and \(-\frac{1}{3}\).
• Finally, the fourth term, -1, results from multiplying 3 by \(-\frac{1}{3}\).

The beauty of geometric sequences lies in this predictable relationship among sequence terms. Each term seamlessly connects to the next, defined by the consistent presence of the common ratio. Understanding how each term is formed is crucial for mastering geometric sequences.

Geometric Progression Basics

**Geometric progression** is a sequence where each term after the first is achieved by multiplying the previous term by a fixed, non-zero number - the common ratio. It's called a 'progression' because it shows how terms successively advance by a multiplicative process.

Key characteristics of geometric progressions include:

• All terms are connected through multiplication.
• The common ratio is the same throughout the sequence.
• The sequence grows (or decreases) exponentially based on the value of the common ratio.

Whether exponential growth or decay, geometric progressions can easily be identified by looking at the consistency of ratios between terms.In a scenario where the terms are 27, -9, 3, and -1, each being \(-\frac{1}{3}\) times the previous term, this constant ratio establishes the sequence as a geometric progression.
Understanding geometric progressions is fundamental in learning how various phenomena, from financial interest calculations to biological growth models, are analyzed and predicted in real-world scenarios.