Problem 10 - College Algebra with Corequisite Support

Question

For the following exercises, determine whether the sequence is geometric. If so, find the common ratio. \(5,5.2,5.4,5.6,5.8, \ldots\)

Step-by-Step Solution

Answer

The sequence is not geometric as the common ratio does not exist.

Step 1: Identify the sequence values

The given sequence is: 5, 5.2, 5.4, 5.6, 5.8, ...

Step 2: Calculate the ratios between terms

To determine if a sequence is geometric, the ratio of consecutive terms must be the same. - Ratio from 5 to 5.2 is: \[ \text{Ratio 1} = \frac{5.2}{5} = 1.04 \]- Ratio from 5.2 to 5.4 is: \[\text{Ratio 2} = \frac{5.4}{5.2} = 1.03846 \] (approximately)- Check Ratio from 5.4 to 5.6: \[\text{Ratio 3} = \frac{5.6}{5.4} = 1.03704 \] (approximately)

Step 3: Compare calculated ratios

Compare the calculated ratios from each consecutive pair of terms: - Ratio 1 is 1.04, - Ratio 2 is approximately 1.03846, - Ratio 3 is approximately 1.03704. Since these ratios are not equal, the sequence is not geometric.

Key Concepts

Common Ratio

A geometric sequence is defined primarily by its **common ratio**. The common ratio is the factor by which you multiply each term to get the next term in the sequence. In mathematical terms, if a sequence is geometric, then the ratio \( r \) between two consecutive terms is consistent throughout the sequence. To find the common ratio, you divide each term by the previous term.

For example, in a geometric sequence with an initial term \( a_1 \), two consecutive terms are typically expressed as \( a_n \) and \( a_{n+1} \). The common ratio \( r \) is calculated as:

• \( r = \frac{a_{n+1}}{a_n} \)

In real-world applications of geometric sequences, understanding the common ratio helps predict future values and understand growth patterns, such as in financial projections or population growth studies.

Consecutive Terms

The **consecutive terms** of a sequence are pairs of numbers that follow one another in order. When evaluating whether a sequence is geometric, the focus is on these consecutive terms since the common ratio is determined from them.

In the exercise provided, the sequence 5, 5.2, 5.4, 5.6, 5.8, each pair of consecutive terms is examined to see if a constant ratio exists. As shown in the solution, dividing the second term by the first term yields a ratio from which you can infer the pattern. Consecutive terms are crucial to identify the nature of the sequence, whether it is geometric or not.

If the common ratio changes between consecutive terms, as it does in the sequence provided, this indicates that the sequence is not geometric.

Sequence Analysis

**Sequence analysis** refers to the process of closely examining the terms of a sequence to determine its characteristics, such as whether it's arithmetic, geometric, or neither. A robust analysis involves calculating the ratios or differences between consecutive terms to check for constancy.

In the provided solution, sequence analysis is crucial. First, the terms are systematically broken down, and calculations are performed to find the ratios between them. Then, by comparing these ratios, it can be determined that the sequence is not geometric because the ratios differ slightly.

Knowing how to analyze a sequence enables you to identify patterns and regularities, which are essential skills in solving problems related to sequences in mathematics. This analysis also guides decision-making in numerous applications like investing or resource allocation where understanding trends is imperative.