Problem 70 - College Algebra with Modeling and Visualization

Question

Determine if \(f\) is a geometric sequence. $$f(n)=-3(0.25)^{n}$$

Step-by-Step Solution

Answer

Yes, \( f(n) = -3(0.25)^n \) is a geometric sequence.

Step 1: Understand the definition of a geometric sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Step 2: Identify the general form of a geometric sequence

A geometric sequence can be expressed in the form: \( a_n = a_1 imes r^{n-1} \), where \( a_1 \) is the first term and \( r \) is the common ratio.

Step 3: Analyze the given function

The function given is \( f(n) = -3(0.25)^n \). This can be compared with the general form: \( a_1 = -3 \) and \( r = 0.25 \).

Step 4: Determine the common ratio

In the function \( f(n) = -3(0.25)^n \), the common ratio \( r \) is 0.25 because each term is obtained by multiplying the previous term by 0.25.

Step 5: Conclude the analysis

Since the function \( f(n) = -3(0.25)^n \) fits the format of a geometric sequence with a common ratio of 0.25, it is indeed a geometric sequence.

Key Concepts

Common Ratio

In any geometric sequence, the common ratio is pivotal. It tells us how each term relates to the one before it. This relationship is constant; every term is obtained by multiplying the previous term by this common ratio. In our example, the function is defined as \( f(n) = -3(0.25)^n \). Here, the common ratio \( r \) is \( 0.25 \).

Understanding the common ratio helps in predicting the sequence's behavior and progression. It's crucial to remember that the ratio must be non-zero. A common ratio of zero would render all subsequent terms in the sequence equal to zero, which doesn't meet the definition of a unique geometric progression.

General Form of Sequence

The general form of a geometric sequence provides a blueprint for understanding or constructing a sequence. It's expressed as \( a_n = a_1 \times r^{n-1} \). This means each term \( a_n \) can be calculated by multiplying the first term \( a_1 \) by the common ratio \( r \), raised to the power of \( n-1 \), where \( n \) is the position of the term in the sequence.

Applying this to our sequence \( f(n) = -3(0.25)^n \), we see that \( a_1 \) (the first term) is \( -3 \), and the common ratio \( r \) is \( 0.25 \). This form is especially useful for identifying whether a sequence is geometric and for determining specific terms or analyzing its overall structure.

Sequence Analysis

Analyzing a sequence involves verifying if it adheres to the properties of the type of sequence it claims to be—in this case, geometric. To conclude that a sequence is geometric, all terms must fit the framework of the general form and common ratio.

In the sequence described by \( f(n) = -3(0.25)^n \), every subsequent term is derived by multiplying the last term by \( 0.25 \), thus satisfying the definition of a geometric sequence.

This analysis not only helps in confirming the type of sequence but also assists in predicting future terms, identifying long-term trends, and easing the computational process in applications of mathematics. Through understanding a sequence's structure, one gains insight into both its repetitive nature and its potential applications in various fields.