Problem 19 - Algebra and Trigonometry

Question

\(19-24\) . Use a graphing calculator to do the following. (a) Find the first 10 terms of the sequence. (b) Graph the first 10 terms of the sequence. $$ a_{n}=4 n+3 $$

Step-by-Step Solution

Answer

The first 10 terms are 7, 11, 15, 19, 23, 27, 31, 35, 39, 43. Plot these points to visualize the sequence.

Step 1: Define the Sequence Formula

The sequence is defined by the formula \( a_n = 4n + 3 \). This means that each term in the sequence is generated by multiplying the term position \( n \) by 4 and then adding 3.

Step 2: Calculate the First 10 Terms

To find the first 10 terms, substitute \( n = 1, 2, 3, ..., 10 \) into the formula \( a_n = 4n + 3 \). Calculate each term individually: \( a_1 = 4 \times 1 + 3 = 7 \), \( a_2 = 4 \times 2 + 3 = 11 \), and so on, up to \( a_{10} = 4 \times 10 + 3 = 43 \).

Step 3: List the First 10 Terms

After performing the calculations, the first 10 terms are: 7, 11, 15, 19, 23, 27, 31, 35, 39, and 43.

Step 4: Graph the First 10 Terms on Calculator

Using a graphing calculator, plot the points from the sequence (1,7), (2,11), (3,15), ..., (10,43) on the grid. Set up the graph with the \( x \)-axis representing the term number and the \( y \)-axis representing the value of the sequence. Ensure each point is correctly placed using the corresponding coordinates.

Step 5: Review the Graph

Once plotted, the graph should show a linear pattern with evenly spaced points, confirming the arithmetic nature of the sequence. The line that could be drawn through all points illustrates the linear increase as defined by the sequence formula \( a_n = 4n + 3 \).

Key Concepts

Sequence Graphing

Graphing a sequence involves plotting each term on a coordinate plane, where the x-axis typically shows the position of the term (n), and the y-axis shows the actual value of each term. For an arithmetic sequence like the one given by the formula \( a_n = 4n + 3 \), every point you graph represents a term in that sequence.

• Start by determining the first 10 terms using the given formula.
• Plot each term as a point on your graphing calculator or graph paper.
• For example, since the first term is 7, you'll plot the point (1, 7).

After plotting all 10 points, you can observe a pattern; these points form a straight line if connected. This visual representation highlights the consistent increase, typical of arithmetic sequences. In this particular instance, each step increases by the same amount, flowering the linear nature of the progression.

Linear Functions

The sequence from the exercise is a great example of an arithmetic sequence, which is closely related to linear functions. The formula \( a_n = 4n + 3 \) is a linear function of n.

• Linear functions are characterized by a constant rate of change, reflected here as "4n," indicating the value changes four units for every increase in n.
• The number 3 represents a vertical shift, starting the graph 3 units above the origin.

The graph of this sequence will display a straight line. The slope of this line directly corresponds to the coefficient of \( n \) in the formula, which is 4. This means for every unit move along the x-axis, the y-value increases by 4. Recognizing these patterns confirms the presence of a linear function within an arithmetic sequence. Linear functions, represented graphically as straight lines, have significant importance in various fields.
Understanding this relationship lays a foundation for comprehending more complex functions in mathematics.

Graphing Calculator Usage

Using a graphing calculator can simplify the visualization and calculation of sequences. Here's how a graphing calculator comes in handy:

• First, input the formula \( a_n = 4n + 3 \) into the calculator.
• Next, use the calculator's table feature to compute values for each term handy. For instance, input \( n = 1 \) through \( n = 10 \) and quickly get corresponding sequence values.
• Graph these points by switching to the graph mode.

Some calculators also offer zoom and scale features, making it easier to see the details of your plotted sequence.
By accurately plotting each term visually, students can see the progression of arithmetic sequences in real-time.
Not only does this method aid in calculation, but it also reinforces understanding of sequence patterns and linear relationships.
This accurate plotting is crucial for students to effectively interpret their mathematical results and better understand the behavior of linear functions in sequence graphing.