Problem 69 - College Algebra with Modeling and Visualization

Question

Determine if \(f\) is a geometric sequence. $$f(n)=4(2)^{n-1}$$

Step-by-Step Solution

Answer

Yes, it is a geometric sequence.

Step 1: Identify the General Form of a Geometric Sequence

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

Step 2: Compare Given Function to General Form

Given is the function \(f(n)=4(2)^{n-1}\). It aligns with the general form \(a_n = a \, r^{(n-1)}\) where \(a = 4\) and \(r = 2\).

Step 3: Identify First Term and Common Ratio

From the function \(f(n)=4(2)^{n-1}\), identify \(a = 4\), and common ratio \(r = 2\).

Step 4: Conclusion about the Nature of Sequence

Since the function can be expressed in the form \(a_n = a \, r^{(n-1)}\), it confirms that **f(n)** is a geometric sequence with first term 4 and common ratio 2.

Key Concepts

Common Ratio

In any geometric sequence, the common ratio is a vital component. It is the constant factor between consecutive terms of the sequence. If you take any term in the sequence, except the first one, and divide it by its preceding term, you'll get the common ratio, denoted by the letter \(r\). For instance, if your sequence begins with 2 and continues as 4, 8, 16, and so on, the common ratio is 2 because \(4 \div 2 = 2\) and \(8 \div 4 = 2\), and this pattern continues.

• The presence of a common ratio is what differentiates a geometric sequence from other sequences.
• It's crucial to determine this ratio to evaluate the behavior of the sequence over time.
• In our given function, \(f(n) = 4(2)^{n-1}\), the common ratio is 2. This illustrates that each term is twice the previous term.

This consistent factor plays an essential role in defining the properties and the expansion of the sequence.

Exponential Function

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. It is an integral part of understanding geometric sequences. In our case, the function \(f(n) = 4(2)^{n-1}\) is an exponential function because the term \(2\) is raised to the power of \(n-1\).

• Exponential functions often model real-world scenarios where quantities grow or decay at a constant relative rate.
• The base of the exponent, in this case, \(2\), influences the rapidity of the sequence growth.
• For this function, it indicates how quickly the terms in our sequence increase.

Exponential functions are pivotal in various fields such as biology, finance, and physics, making them highly applicable in both academic and practical realms. They help us predict phenomena such as population growth, radioactive decay, and compound interest.

Sequence and Series

Sequences and series are fundamental concepts in mathematics that describe ordered sets of numbers and their cumulative sums, respectively. A sequence is essentially a list of numbers following a specific pattern, like our geometric sequence \(4, 8, 16, 32, \ldots\) generated by the function \(f(n) = 4(2)^{n-1}\).

• A series, meanwhile, refers to the sum of the terms in a sequence.
• For instance, when you add up the terms: \(4 + 8 + 16 + 32 + \cdots\), it forms a series.
• Understanding the sequence helps comprehend the series as it allows one to predict the summation behavior over several terms.

Both sequences and series are integral in calculus and algebra, forming the basis for analyzing complex patterns and solving various mathematical problems. They provide insights into the nature of the numbers involved and the relationships between them.