Problem 48 - 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.) $$a_{n}=-5+2 n$$

Step-by-Step Solution

Answer

Generate the first 10 terms of the sequence using the given formula, represent each term as a point on the graph, and draw the graph by connecting these points.

Step 1: Compute the sequence terms

Start with \(n=1\) and substitute into the sequence formula. Continue calculating for \(n=2\) through \(n=10\).

Step 2: Create a Coordinate Pair

Each term of the sequence corresponds to an \(x-y\) pair that can be plotted on a graph. We can represent each term of the sequence as a coordinate pair \((n, a_{n})\), where \(n\) is from 1 to 10.

Step 3: Draw the Graph

Using a graphing utility, plot each coordinate pair and connect the dots to form the graph.

Key Concepts

Graphing Utility

Using a graphing utility is one of the most effective methods for visualizing mathematical concepts, especially sequences. This digital tool allows students to plot functions and sequences quickly, providing immediate visual feedback. When dealing with sequences, it's common to plot the term number (also known as the index) on the x-axis and the term value on the y-axis.

For the exercise at hand, the graphing utility streamlines the plotting process for the first ten terms of an arithmetic sequence. Once the values are calculated, they can be immediately entered into the graphing utility. By observing the plotted points and the pattern they follow, students can gain a deeper understanding of the sequence's behavior and predict future terms.

For optimal understanding, students should ensure they are familiar with how to input sequences into the graphing utility, scale their axes for clear visualization, and alter settings to best represent the sequence on their screen.

Arithmetic Sequence

An arithmetic sequence is a list of numbers with a constant difference between consecutive terms. This difference is known as the common difference, and it plays a crucial role in the properties and behavior of the sequence. In the exercise provided, the sequence is given by the formula \(a_{n}=-5+2n\). The term \(a_{n}\) represents the nth term of the sequence, and the common difference here is 2 since each succeeding term increases by 2.

To understand the arithmetic sequence, students should learn how to derive the formula for the nth term, which in this case is a linear function of \(n\). Knowing this allows students to compute any term in the sequence without the need to know all the previous terms. It's also valuable for students to recognize the graphical representation of an arithmetic sequence, which will be a set of points that lie in a straight line when plotted.

Coordinate Pairs

Coordinate pairs are a foundational concept in graphing, providing a method to represent points on a two-dimensional plane. Each point is defined by an x-coordinate (horizontal placement) and a y-coordinate (vertical placement). In the context of sequences, each term can be expressed as a coordinate pair, where the term number is the x-coordinate and the term value is the y-coordinate.

For the given arithmetic sequence, you would express the first term as \( (1, a_{1}) \) and continue this process up to the tenth term, \( (10, a_{10}) \). When these coordinate pairs are plotted on a Cartesian plane and connected, they should form a clear, linear pattern due to the nature of the arithmetic sequence involved. Understanding how to interpret and plot these points is crucial for graphically representing various mathematical relationships, including sequences and functions.