Problem 72 - College Algebra with Modeling and Visualization

Question

Determine if \(f\) is a geometric sequence. $$f(n)=2(n-1)^{n}$$

Step-by-Step Solution

Answer

No, \(f(n) = 2(n-1)^n\) is not a geometric sequence.

Step 1: Confirm the General Form of a Geometric Sequence

A geometric sequence is given by \(a_n = a_1 \cdot r^{n-1}\), where \(a_1\) is the first term and \(r\) is the common ratio. Our task is to find these components in the given function \(f(n) = 2(n-1)^n\).

Step 2: Identify Terms of the Sequence

Compute the first few terms of the sequence by substituting values of \(n\) into the function.\[\begin{align*}f(1) &= 2(1-1)^1 = 0, \f(2) &= 2(2-1)^2 = 2, \f(3) &= 2(3-1)^3 = 16, \f(4) &= 2(4-1)^4 = 162. \end{align*}\]

Step 3: Calculate Ratios to Determine Common Ratio

The common ratio \(r\) for a geometric sequence is found by dividing consecutive terms: \[\begin{align*}\text{Ratio 1 (between }f(2) \text{ and } f(1)\text{)} &= \frac{2}{0}, \text{ undefined}, \\text{Ratio 2 (between }f(3) \text{ and } f(2)\text{)} &= \frac{16}{2} = 8, \\text{Ratio 3 (between }f(4) \text{ and } f(3)\text{)} &= \frac{162}{16} = 10.125. \end{align*}\]

Step 4: Analyze the Results

The calculated ratios are not consistent; specifically, the first ratio is undefined, and the remaining two ratios differ (8 and 10.125). In a geometric sequence, the ratios must be the same for all consecutive terms. Hence, \(f(n)\) does not form a geometric sequence.

Key Concepts

Common Ratio

In the world of geometric sequences, the common ratio is a crucial element. It tells us the factor by which we multiply to get from one term in the sequence to the next. For a sequence to be geometric, all consecutive terms must be related by a constant factor, known as the common ratio, denoted as \(r\).

The general form used to express a geometric sequence involves this common ratio: \(a_n = a_1 \cdot r^{n-1}\), where \(a_1\) is the first term and \(a_n\) represents any term in the sequence.

In our specific case, we attempted to determine if \(f(n) = 2(n-1)^n\) was geometric by calculating ratios between consecutive terms. However, the presence of undefined or inconsistent ratios (0/2, 8, and 10.125) revealed that no such constant common ratio exists for this function. Without a constant \(r\), we do not have a geometric sequence.

Terms of a Sequence

Let's dive into what terms of a sequence really mean and how they are derived, especially in the context of geometric sequences. Each term in a sequence is derived using a specific formula, which allows you to compute the values resulting in the sequence.

For\(f(n) = 2(n-1)^n\), we calculate terms by substituting integer values into \(n\). For example:

• \(f(1) = 2(1-1)^1 = 0\)
• \(f(2) = 2(2-1)^2 = 2\)
• \(f(3) = 2(3-1)^3 = 16\)
• \(f(4) = 2(4-1)^4 = 162\)

Each term provides insight into understanding if the function follows a specific type of sequence, such as geometric. However, without consistent ratios between terms, a sequence cannot be classified as geometric.

Algebraic Functions

Understanding algebraic functions is key to determining the nature of sequences. An algebraic function is a mathematical expression built out of algebraic operations which include addition, subtraction, multiplication, division, and exponentiation.

In the context of the given problem, we explored the function \(f(n) = 2(n-1)^n\). This function empowers us to construct a sequence by inputting values of \(n\), and utilizing algebraic operations to derive various terms. It's important to translate these expressions to understand the behavior of the sequence.

While algebraic functions can describe various sequence types, they don't automatically indicate if a sequence is arithmetic, geometric, or otherwise. It is through analysis, such as attempting to find a common ratio, that we identify the specific nature of the sequence. In this problem, despite the presence of algebraic expressions, the lack of consistent ratios indicates the absence of a geometric nature.