Problem 22 - 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. $$\frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \frac{1}{8}, \dots$$

Step-by-Step Solution

Answer

The sequence is not geometric; the consecutive ratios are not equal.

Step 1: Identify Consecutive Ratios

To determine if the sequence is geometric, find the ratios of each consecutive pair of terms. Compute the ratio between the second term \( \frac{1}{4} \) and the first term \( \frac{1}{2} \), then between the third term \( \frac{1}{6} \) and the second term \( \frac{1}{4} \), and finally between the fourth term \( \frac{1}{8} \) and the third term \( \frac{1}{6} \).

Step 2: Compute First Ratio

The ratio between the second term \( \frac{1}{4} \) and the first term \( \frac{1}{2} \) is calculated as follows: \[ \frac{ \frac{1}{4} }{ \frac{1}{2} } = \frac{1}{4} \times \frac{2}{1} = \frac{2}{4} = \frac{1}{2} \]

Step 3: Compute Second Ratio

The ratio between the third term \( \frac{1}{6} \) and the second term \( \frac{1}{4} \) is: \[ \frac{ \frac{1}{6} }{ \frac{1}{4} } = \frac{1}{6} \times \frac{4}{1} = \frac{4}{6} = \frac{2}{3} \]

Step 4: Compute Third Ratio

The ratio between the fourth term \( \frac{1}{8} \) and the third term \( \frac{1}{6} \) is: \[ \frac{ \frac{1}{8} }{ \frac{1}{6} } = \frac{1}{8} \times \frac{6}{1} = \frac{6}{8} = \frac{3}{4} \]

Step 5: Analyze the Ratios

For a sequence to be geometric, all consecutive ratios must be equal. The computed ratios are \( \frac{1}{2} \), \( \frac{2}{3} \), and \( \frac{3}{4} \). Since these are not equal, the sequence is not geometric.

Key Concepts

Sequence

A sequence is a list of numbers arranged in a specific order. Each number in the sequence is called a term. Sequences can have different patterns or rules that define the relationship between terms. There are various types of sequences such as arithmetic, geometric, and others. In an arithmetic sequence, the difference between consecutive terms is constant.
In contrast, in a geometric sequence, the ratio between consecutive terms remains constant. Sequences are integral in mathematics as they help in understanding patterns, predicting future terms, and solving complex problems. It is important to analyze the sequence's structure to determine which type it might be.

Common Ratio

In a geometric sequence, understanding the common ratio is crucial. It is this constant number that links each term to the next. To find the common ratio, simply divide any term by the previous one in the sequence.

• The presence of a common ratio signifies that the sequence is geometric.
• All terms will be multiplied by this constant ratio to produce the subsequent term.

The common ratio can be a fraction, a whole number, or even a positive or negative number. Recognizing it helps in predicting other terms in the sequence without directly computing each consecutive term.

Consecutive Terms

Consecutive terms in a sequence are terms that follow one right after the other. They help identify the pattern in the sequence, especially when trying to determine if it is geometric.

• By comparing consecutive terms, we can calculate and verify the common ratio.
• In a geometric sequence, all such consecutive ratios should be the same for the sequence to be truly geometric.

Consistently analyzing consecutive terms allows us to uncover the underlying nature of the sequence and conclude whether it holds the characteristics of a geometric progression.

Ratio Calculation

Ratio calculation is a mathematical method used to compare two numbers. In the context of sequences, it involves dividing one term by its predecessor. This calculation is at the heart of determining if a sequence is geometric.
When calculating the ratio between consecutive terms, at each step, one evaluates whether these ratios are constant throughout.

• If they are, the sequence is geometric.
• If not, the sequence is not a geometric series.

By practicing ratio calculation, students can classify sequences more effectively and predict future terms in a geometric sequence.