Question

Compare the graph of \(a_n=5(3)^{n-1}\), where \(n\) is a positive integer, to the graph of \(f(x)=5 \cdot 3^{x-1}\), where \(x\) is a real number.

Short answer

The graph of sequence \(a_n = 5(3)^{n-1}\) will consist of distinct points and the graph of function \(f(x) = 5 \cdot 3^{x-1}\) will be a smooth curve. They share the same y-values for integer x-values, so the points of \(a_n\) lie on the curve of \(f(x)\).

Step-by-step solution

Explain the sequence

The sequence \(a_n = 5(3)^{n-1}\) can be only defined for positive integer values of \(n\). This means the graph of the sequence will consist of individual points.

Explain the function

The function \(f(x) = 5 \cdot 3^{x-1}\) is defined for any real number \(x\). It implies that the graph of the function will be a smooth curve.

Comparison of the graphs

Both the sequence and the function will have the same y-values for integer values of \(x\) or \(n\). This is because \(f(x) = 5 \cdot 3^{x-1}\) and \(a_n = 5(3)^{n-1}\) have the same calculation rule for these values. The difference lies in the values of \(x\) that are not integers in \(f(x)\). The sequence \(a_n\) won't have values here, but there will be values on \(f(x)\). So \(f(x)\) will appear as a smooth curve while \(a_n\) will be a collection of distinct points along that curve.

Key concepts

Algebraic Sequences

An algebraic sequence, frequently encountered in mathematical studies, is simply a list of numbers generated by following a rule or a formula. The sequence mentioned in the exercise, (\(a_n = 5(3)^{n-1}\), showcases a specific type of algebraic sequence called a geometric sequence. In this sequence, each term is determined by multiplying the previous term by a constant value, in this case, 3.
Geometric sequences have wide applications, including in finance to model exponential growth or decay. Understanding how to graph these sequences is an essential skill. Algebraic sequences are discrete, defined only for positive integers. That's why its graph consists of distinct points rather than a continuous line, highlighting its principal characteristic: it describes a scenario where observations are made at specific intervals rather than continuously. For example, compound interest calculated at fixed time intervals can be modeled using algebraic sequences.

Exponential Functions

Exponential functions, like (\(f(x) = 5 \times 3^{x-1}\), are fundamental in both mathematics and real-world applications, since they describe continuous growth or decay processes. Unlike algebraic sequences, exponential functions are continuous and defined for all real numbers. The base of the exponential function, in this case, the number 3, indicates the rate of growth when it's greater than one, or decay when it's between zero and one.
Students can spot the differences by observing how the graph of an exponential function doesn't have gaps—as it includes all possible values within the domain of real numbers. These functions are used in various fields such as population studies, radioactive decay, and interest calculations that are compounded continuously. Recognizing the behavior of exponential functions is vital, especially their rapid increase or decrease, which calls for careful consideration when modeling real-life situations.

Graphing Sequences and Functions

Understanding how to effectively graph sequences and functions can provide intuitive insight into their behavior. When graphing the geometric sequence (\(a_n = 5(3)^{n-1}\), you place dots at the integer values of n. These individual points reflect the nature of sequences, which are based on distinct occurrences. The process of graphing functions like (\(f(x) = 5 \times 3^{x-1}\) involves plotting a smooth curve that continuously flows across the graph without interruption, showing all real number inputs.
By graphing both the sequence and the function on the same set of axes, students can see exactly where the sequence 'sits' on the function. This visual representation helps to solidify the understanding of how discrete and continuous models are related to each other and how they differ. Generally, graphing offers a practical way to compare different mathematical concepts and to explore the behavior of sequences and functions graphically.

Real Number System

The real number system comprises both rational numbers (which can be expressed as fractions) and irrational numbers (which cannot be expressed as fractions and have nonrepeating, nonterminating decimal expansions). Therefore, any value along the continuum of real numbers can be an input to (\(f(x) = 5 \times 3^{x-1}\), the function in our exercise. This is not the case for the sequence, which is confined to integer inputs.
A firm understanding of the real number system is crucial when dealing with functions such as exponentials. It explains why the graph of an exponential function contains all the points for any real number you can think of, while sequences are limited to particular points that correspond to the integers within the real numbers. The intricacy of the real number system is what allows continuous functions to model an incredibly diverse range of phenomena, contrasting with the specificity of sequences which are used for models based explicitly on discrete values.