Problem 17 - Algebra and Trigonometry

Question

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, and 43.

Step 1: Understand the Sequence Formula

The sequence is given by the formula \(a_{n} = 4n + 3\). This means that the \(n\)-th term of the sequence can be calculated by multiplying \(n\) by 4 and then adding 3.

Step 2: Calculate the First Term

Substitute \(n = 1\) into the formula: \(a_{1} = 4(1) + 3 = 7\). So, the first term is 7.

Step 3: Calculate the Second Term

Substitute \(n = 2\) into the formula: \(a_{2} = 4(2) + 3 = 11\). Hence, the second term is 11.

Step 4: Continue Calculating Terms

Repeat the process for \(n = 3, 4, \, \ldots , \, 10\) to calculate each term. Here are the results:- \(a_{3} = 4(3) + 3 = 15\)- \(a_{4} = 4(4) + 3 = 19\)- \(a_{5} = 4(5) + 3 = 23\)- \(a_{6} = 4(6) + 3 = 27\)- \(a_{7} = 4(7) + 3 = 31\)- \(a_{8} = 4(8) + 3 = 35\)- \(a_{9} = 4(9) + 3 = 39\)- \(a_{10} = 4(10) + 3 = 43\)

Step 5: Graph the Sequence

Use a graphing calculator to plot the sequence terms. Enter the pairs \((n, a_n)\) for \(n = 1\) to 10 as coordinates. Plot the points on a coordinate plane where the x-axis represents \(n\) and the y-axis represents the sequence value \(a_n\). Connect the dots if needed to visualize the linear relationship.

Key Concepts

Graphing Calculator

A graphing calculator is a powerful tool that helps visualize mathematical functions and sequences. To use it for graphing sequences, you'll need to enter your data correctly.
Begin by switching the calculator to sequence mode. This allows you to enter each term of the sequence directly. Key in the formula you've been given, such as \(a_{n} = 4n + 3\).
After entering the formula, use the graph function to display the sequence visually. The calculator takes your sequence formula and generates a set of points that show how the sequence progresses. This visual representation can make understanding sequences much simpler.

Sequence Formula

The sequence formula \(a_{n} = 4n + 3\) is a type of arithmetic sequence. Arithmetic sequences are simple patterns where you add the same number each time to get the next term.
In this case, the term \(a_{n}\) is calculated by multiplying the term number \(n\) by 4 and then adding 3. The number 4 here is called the common difference since it's what we add as we move from one term to the next.

• Common difference: 4
• Starting value when \(n = 1\): 7

By understanding this formula, you can easily calculate any term in the sequence without having to find all the previous terms.

Linear Equation

A linear equation in its simplest form looks like \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. The sequence formula \(a_{n} = 4n + 3\) is essentially a linear equation written with different variables.
In this formula, 4 is the slope, showing how steeply the sequence increases. The 3 acts like the y-intercept, indicating where the line crosses the y-axis in the context of a graph.

• The formula is linear because the graph of its terms forms a straight line.
• Every increase in \(n\) directly corresponds to an increase in \(a_{n}\) by 4, showing constant growth.

Understanding it as a linear equation helps you anticipate the overall trend of the sequence without computing each term.

Graphing Sequences

Plotting sequences on a graph can provide a clear picture of the sequence's behavior. When you graph \(a_{n} = 4n + 3\), you're actually plotting points on a coordinate plane.
Here, \(n\) represents the x-coordinate, and \(a_{n}\) is the y-coordinate. Each point corresponds to a term in the sequence, like (1, 7), (2, 11), and so on. Once you plot these points, you’ll notice they lie along a straight line. This visual cue emphasizes the sequence's linear nature, simplifying analysis.
When using a graphing calculator, connecting these points can help visualize the sequence as a continuous line, highlighting the sequence's predictable increase.