Problem 85 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the first 10 terms of the sequence. (Assume \(n\) begins with 1.) $$a_{n}=\frac{2 n}{n+1}$$

Step-by-Step Solution

Answer

The graph will show a decreasing trend as the values of \(n\) increase. The initial term (n=1) will start near 2 and the term (n=10) will end near 1.6.

Step 1: Calculate Sequence Terms

Substitute values of \(n\) from 1 to 10 into the sequence formula \(a_{n} = \frac{2n}{n+1}\) to find the sequence terms. This will give an array of the calculated sequence terms: \[a_1, a_2, a_3, ..., a_{10} \], where each \(a_i = \frac{2*i}{i+1}\) for \(i = 1, 2, ..., 10\)

Step 2: Plot Sequence Terms

Now that the 10 terms are determined, they can be plotted on a graph. The horizontal axis will represent the sequence terms (n values, with n = 1 to 10), and the vertical axis will represent the values of the sequence terms. Each point on the graph is determined by the coordinates (n, \(a_{n}\)).

Key Concepts

Sequence Graphing Utility

When studying sequences, a sequence graphing utility can be an invaluable tool for visualizing the progression of sequence terms. It essentially transforms a numerical, sometimes abstract algebraic expression into a graphic representation, making it easier to understand and analyze the sequence's behavior. Students use such utilities to input their sequence formula, for example, the given formula of our exercise \(a_{n}=\frac{2n}{n+1}\), and instantly see how the sequence evolves as values of \(n\) increase.

Modern graphing utilities can often handle a variety of sequences, including arithmetic, geometric, and more complex algebraic sequences. They allow users to adjust the view, zoom in on areas of interest, and often provide additional features like generating a table of values alongside the graph. By graphing the first 10 terms, as stipulated in the exercise, students can understand such concepts as convergence, limits, and the overall growth or decay rate of a sequence.

Plotting Sequence Terms

The process of plotting sequence terms turns numeric data into a visual format that is often more digestible. In the context of sequences, this involves placing points on a Cartesian coordinate system where each point represents a term from the sequence. To plot the terms of the sequence \(a_{n}=\frac{2n}{n+1}\) mentioned in the exercise, you would begin by calculating the sequence terms for values of \(n\) from 1 to 10.

Once calculated, each term denotes a specific point whose coordinates are \(\left(n, a_{n}\right)\). For the given exercise, this would result in plotting points such as (1, \(a_{1}\)), (2, \(a_{2}\)), and so on, up to (10, \(a_{10}\)). Plotting these points can reveal trends, such as how quickly the sequence values approach a certain limit, or if they fluctuate over time. This visual representation complements the numerical analysis and provides a fuller understanding of the sequence's characteristics.

Algebraic Sequences

An important subset of sequences in mathematics is that of algebraic sequences, which includes the sequence in our exercise. These sequences are characterized by an algebraic expression with variables and coefficients that determine the progression of the sequence. The sequence \(a_{n}=\frac{2n}{n+1}\), for example, is defined by a rational expression involving a simple polynomial in the numerator and denominator.

Each term in an algebraic sequence is generated by plugging successive integers into the algebraic formula. For instance, when \(n=1\), the first term \(a_{1}\) is \(2/2\), when \(n=2\), the second term \(a_{2}\) is \(4/3\), and so on. These terms often reveal patterns of growth or behaviors as \(n\) increases – some may approach a fixed number (convergence), some may continue to grow or decrease without bound, and others may oscillate. Understanding algebraic sequences is crucial as they are foundational in higher mathematics, being used to describe complex systems and model real-world phenomena.