Problem 60 - College Algebra with Corequisite Support

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 points for first 5 terms: (1, -7), (2, -2), (3, 3), (4, 8), (5, 13).

Step 1: Identify Sequence Formula

The formula for the arithmetic sequence is given as \(a_n = -12 + 5n\), where \(n\) is the term number. This means for each term, you multiply the term number by 5 and then subtract 12 to get the term's value.

Step 2: Calculate the First Term

To find the first term, substitute \(n=1\) into the formula: \(a_1 = -12 + 5 \times 1 = -7\).

Step 3: Calculate the Second Term

Substitute \(n=2\) into the formula: \(a_2 = -12 + 5 \times 2 = -2\).

Step 4: Calculate the Third Term

Substitute \(n=3\) into the formula: \(a_3 = -12 + 5 \times 3 = 3\).

Step 5: Calculate the Fourth Term

Substitute \(n=4\) into the formula: \(a_4 = -12 + 5 \times 4 = 8\).

Step 6: Calculate the Fifth Term

Substitute \(n=5\) into the formula: \(a_5 = -12 + 5 \times 5 = 13\).

Step 7: Plot the Points

Now, plot the points (1, -7), (2, -2), (3, 3), (4, 8), and (5, 13) on a graph where the x-axis represents the term number \(n\) and the y-axis represents the term value \(a_n\).

Step 8: Draw the Graph

Draw a straight line through the points plotted to represent the arithmetic sequence, as the differences between consecutive terms are constant.

Key Concepts

Term Calculation in Arithmetic Sequences

Arithmetic sequences are a type of sequence in mathematics where each term differs from the previous one by a constant amount. This constant difference is known as the 'common difference'. In the given sequence formula, \(a_n = -12 + 5n\), the common difference is 5. This is the amount added to each term to get to the next one.

Calculating terms in an arithmetic sequence involves simply plugging a sequential number into the position of \(n\) in the sequence formula. Here's how the term calculation works:

• For the first term \(a_1\), substitute 1 for \(n\): \(a_1 = -12 + 5 \times 1 = -7\).
• For the second term \(a_2\), substitute 2: \(a_2 = -12 + 5 \times 2 = -2\).
• Continue this pattern by increasing \(n\) to get subsequent terms: \(a_3 = 3\), \(a_4 = 8\), and \(a_5 = 13\).

By calculating a few terms, you can begin to see and understand the pattern of the sequence. This method provides a straightforward way to explore the behavior and structure of arithmetic sequences.

Sequence Graphing

Graphing an arithmetic sequence helps visualize the relationship between the term number and its value. Each term number, \(n\), is plotted on the x-axis. Meanwhile, the calculated value of the term, \(a_n\), is plotted on the y-axis. This forms a series of points that can be connected.

For our sequence, the points to plot are derived from the calculated terms: (1, -7), (2, -2), (3, 3), (4, 8), and (5, 13). Plot these points:

• (1, -7): First term is positioned at x=1 and y=-7.
• (2, -2): Second term at x=2 and y=-2, indicating a rise from the first term.
• (3, 3): Third term continues the upward trend.
• (4, 8) and (5, 13): Subsequent terms carry on this pattern.

By connecting these points with a straight line, the entire sequence is visualized as a straight line on the graph. This visual representation indicates the uniform growth that characterizes arithmetic sequences.

Linear Equation Representation

An arithmetic sequence can be understood as a linear equation, a straight line on a graph when plotted. The equation \(a_n = -12 + 5n\) adheres to the familiar formula for a linear equation, \(y = mx + c\), where \(m\) represents the slope, and \(c\) represents the y-intercept.

In the context of our sequence:

• The slope, \(m = 5\), shows how much the sequence increases by each step.
• The y-intercept, \(c = -12\), indicates where the sequence would intersect the y-axis if \(n\) were to equal zero.

This linear representation confirms the constant difference in the sequence. Understanding arithmetic sequences as linear equations allows easy prediction of terms even without direct calculation, since the relationship is predictable and consistent.