Problem 49 - 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.) $$a_{n}=0.4 n-2$$

Step-by-Step Solution

Answer

To solve this, generate the 10 terms of the sequence by substituting values of n from 1 to 10. Plot the terms using a graphing utility, which will result in 10 distinct points on the graph.

Step 1: Identify the sequence

The given sequence is \( a_n = 0.4n - 2 \), where n represents the term number, starting from 1. It's a linear sequence.

Step 2: Generate the first 10 terms

Substitute values of \( n \) from 1 to 10 into the sequence to get the first 10 terms. This will create 10 ordered pairs (n , \( a_n \)) to use for graphing.

Step 3: Plot the sequence

Use a graphing utility and plot the calculated 10 ordered pairs on the graph. As this is a sequence, not a function, it will be graphed as distinct points, not a continuous line.

Key Concepts

Linear Sequences

When studying sequences in mathematics, a linear sequence is a series of numbers in which the difference between consecutive terms is constant. In other words, if you subtract any term from the subsequent term, you'll always get the same result. This constant difference is also known as the 'common difference'.

Consider the linear sequence defined by the formula ( a_n = 0.4n - 2 ). Here, 'n' represents the position of the term in the sequence. For each increase by 1 in 'n', the term '0.4n' increases by 0.4, which shows the pattern of growth that characterizes a linear sequence. This kind of predictable pattern is what makes linear sequences particularly straightforward to work with - once you find the common difference, generating terms becomes a matter of simple arithmetic.

Constructing a graph representing a linear sequence helps visualize this constant growth rate. When graphed, the terms form a set of points that align to create the shape reminiscent of a straight line - hence the term 'linear'. It's important to note that despite this, the sequence itself is discrete, consisting of clear and separate values, not a continuous line.

Graphing Utility

A graphing utility is an essential tool for visualizing mathematical concepts, especially sequences. This can be a software program, an application, or an advanced calculator capable of plotting graphs based on numerical data. When it comes to sequences, graphing utilities allow us to plot the individual terms as points, providing a visual representation of the sequence's behavior.

These utilities are incredibly user-friendly and usually require just basic inputs – in this case, the ordered pairs that represent the terms of a sequence. After these inputs are provided, the graphing utility processes them and produces a visual graph. The advantage of using such a tool is its accuracy and the ability to visualize complex patterns that might be difficult to interpret through numbers alone.

Especially for educational purposes, the graphical representation can make abstract concepts more concrete, helping students to understand the progression and relationship between the terms of a sequence better. And with the increasing integration of technology in education, such utilities are widely accessible and easy to use, often included in mathematics learning platforms or available as standalone apps.

Ordered Pairs

In graphing, an ordered pair is a pair of numbers used to represent the coordinates of a point on a plane. The first number (the x-coordinate) indicates horizontal position, and the second number (the y-coordinate) indicates vertical position. In the context of sequences, each term can be turned into an ordered pair, with 'n' representing the position in the sequence becoming the x-coordinate, and the actual value of the term becoming the y-coordinate.

For example, if we refer to the given sequence ( a_n = 0.4n - 2 ), and we wish to find the ordered pair for the third term where n equals 3, we substitute 3 in place of 'n' to find the y-coordinate:( a_3 = 0.4*3 - 2 ) which simplifies to ( a_3 = 1.2 - 2 ), finally giving us ( a_3 = -0.8 ). Therefore, the third term of our sequence can be represented by the ordered pair (3, -0.8).

Once several ordered pairs from a sequence are determined, they can be plotted on a coordinate grid. If it's a linear sequence, we expect these points, when connected, to suggest a line with a uniform gradient indicated by the sequence's common difference. The technique of converting terms to ordered pairs and then graphing them is integral to understanding the sequence's properties and behavior.