Problem 75 - College Algebra with Modeling and Visualization

Question

Determine if \(f\) is a geometric sequence. \(\begin{array}{rrrrrr}n & 1 & 2 & 3 & 4 & 5 \\ f(n) & \frac{1}{2} & \frac{3}{4} & 1 & \frac{5}{4} & \frac{5}{2}\end{array}\)

Step-by-Step Solution

Answer

The sequence \(f(n)\) is not a geometric sequence.

Step 1: Understand a Geometric Sequence

A geometric sequence is a sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (r). The formula for the nth term of a geometric sequence is given by \(a \cdot r^{(n-1)}\), where \(a\) is the first term and \(n\) is the term number.

Step 2: Calculate Potential Common Ratios

To determine if the sequence given by \(f(n)\) is geometric, calculate the ratio \(\frac{f(n+1)}{f(n)}\) for consecutive terms: 1. \(r_1 = \frac{f(2)}{f(1)} = \frac{\frac{3}{4}}{\frac{1}{2}} = \frac{3}{2}\)2. \(r_2 = \frac{f(3)}{f(2)} = \frac{1}{\frac{3}{4}} = \frac{4}{3}\)3. \(r_3 = \frac{f(4)}{f(3)} = \frac{\frac{5}{4}}{1} = \frac{5}{4}\)4. \(r_4 = \frac{f(5)}{f(4)} = \frac{\frac{5}{2}}{\frac{5}{4}} = 2\).

Step 3: Compare the Ratios

To confirm if the sequence is geometric, check if all calculated ratios \(r_1, r_2, r_3, \) and \(r_4\) are the same. Since the calculated ratios are \(\frac{3}{2}, \frac{4}{3}, \frac{5}{4},\) and \(2\), which are not equal, the sequence is not geometric.

Key Concepts

Common Ratio

In a geometric sequence, every term after the first is determined by multiplying the previous term with a constant, known as the common ratio. This ratio is crucial because it dictates the entire sequence's progression.
To find this common ratio, denoted as \(r\), you need to divide a term by its preceding term. In general, if you have a sequence \(a_1, a_2, a_3, \ldots\), the common ratio \(r\) is given by \(r = \frac{a_{n+1}}{a_n}\).
This ratio remains the same throughout the sequence. In our exercise, however, the calculated ratios \(\frac{3}{2}, \frac{4}{3}, \frac{5}{4}\), and \(2\) differ, confirming that common ratio consistency is absent here. This inconsistency means that the given sequence is not geometric.

Nth Term

The nth term of a geometric sequence allows us to identify any term's value given its position \(n\) without needing to compute all prior terms. The formula is typically expressed as \(a \cdot r^{(n-1)}\), where \(a\) is the first term of the sequence and \(r\) is the common ratio.
Understanding this formula is key because it demonstrates how a geometric sequence expands or contracts exponentially, depending on whether \(|r| > 1\) or \(|r| < 1\). For sequences with \(r = 1\), each term remains the same, and with \(r = -1\), they oscillate.
However, as demonstrated by our exercise where there is no uniform \(r\), we cannot apply this formula effectively, which again shows the sequence isn't geometric.

Sequence Analysis

Analyzing a sequence often involves determining patterns and consistency throughout, crucial for identifying whether it's geometric. Typically, you'll perform sequence analysis by examining the relationship between consecutive terms. If each term is obtained by multiplying the previous one by the same factor (common ratio), the sequence is geometric.
This analysis checks the key property: identical ratios. In our exercise, this verification step led to realizing the sequence is not geometric, due to varying calculated ratios.
Knowing how and when to apply such analysis is essential. It empowers you to classify sequences correctly, discern patterns, and apply the right mathematical tools for further exploration. Each step informs whether a formula like the nth term formula can be effectively used, or if another approach is needed.