Problem 60 - College Algebra

Question

For the following exercises, use the information provided to graph the first 5 terms of the arithmetic sequence. $$ a_{n}=-12+5 n $$

Step-by-Step Solution

Answer

Plot the points (-7, 1), (-2, 2), (3, 3), (8, 4), (13, 5) and connect them with a straight line.

Step 1: Understand the Arithmetic Sequence Formula

The given formula of the arithmetic sequence is \(a_{n} = -12 + 5n\). Here, \(n\) represents the term number in the sequence, and the expression on the right calculates the value of that term. It's important to recognize that this sequence has a starting value (when \(n = 0\)) and a consistent difference of 5 between consecutive terms.

Step 2: Calculate the First 5 Terms

To find the first 5 terms of the sequence, substitute \(n = 1, 2, 3, 4, 5\) into the formula and calculate each term:- When \(n = 1\), \(a_1 = -12 + 5(1) = -7\).- When \(n = 2\), \(a_2 = -12 + 5(2) = -2\).- When \(n = 3\), \(a_3 = -12 + 5(3) = 3\).- When \(n = 4\), \(a_4 = -12 + 5(4) = 8\).- When \(n = 5\), \(a_5 = -12 + 5(5) = 13\).

Step 3: Plot the Points

Using the terms calculated in Step 2, plot the points \(1, -7\), \(2, -2\), \(3, 3\), \(4, 8\), and \(5, 13\) on a Cartesian plane. Treat each point as a coordinate where the first value corresponds to the term number \(n\) and the second value is the sequence value \(a_n\).

Step 4: Connect the Dots

After plotting the points, connect them with a straight line. Since this is an arithmetic sequence with a constant difference (common difference), the points should form a straight line.

Key Concepts

Graphing Sequences

Graphing sequences like an arithmetic sequence helps us visualize how the sequence evolves as we move through its terms. For the sequence given by the formula \(a_{n} = -12 + 5n\), each term corresponds to a specific input in a graph. This exercise involves plotting the sequence's terms on a two-dimensional plane, which typically involves:

• Using the x-axis to represent the term number \(n\).
• Using the y-axis to represent the value of each term \(a_n\).

Graphing allows you to see the trend of the sequence at a glance, witnessing the linearity due to the constant addition or subtraction, known as the common difference. More complex or non-linear sequences would not create a straight line, showcasing the power of graphing in identifying sequence characteristics.

Plotting Points

Plotting points involves converting the terms of the arithmetic sequence into visual coordinates. For this specific sequence:

• Calculate each term's value using \(n\).
• Transform these values into coordinates in the form \((n, a_n)\).

For example, if we determine that when \(n = 1\), \(a_1 = -7\), then the corresponding point is \((1, -7)\). Once you have all your points, typically for the first few terms of the sequence, they can be plotted on a Cartesian plane. This helps in easily identifying the relationship between the term numbers and their respective values. When done correctly, you will see the characteristic straight line form due to the arithmetic sequence.

Common Difference

The common difference in an arithmetic sequence is what distinguishes it from other types of sequences. It represents the constant value added (or subtracted) from one term to the next.

• In the sequence formula \(a_{n} = -12 + 5n\), the common difference is \(5\).
• It indicates that each consecutive term increases by \(5\) units.

This uniform change permits the formation of a straight line when plotting these sequences. Recognizing the common difference is essential for identifying patterns and predicting future terms. This consistency makes mathematical predictions and algebraic expressions easier to handle and verify.

Algebraic Expressions

An algebraic expression in the context of sequences is a way to describe the relationship between the term number and the term's value. For the arithmetic sequence \(a_{n} = -12 + 5n\):

• The expression consists of a starting point, \(-12\), representing the value when \(n = 0\).
• The term \(5n\) illustrates how each increment in \(n\) changes the sequence by the common difference.

Algebraic expressions are compact forms that allow quick calculations of the sequence's terms without having to calculate each term individually. They are essential in understanding sequences as they provide a clear model of how sequences behave mathematically, making them powerful tools for predicting and analyzing sequences.