Problem 21 - Precalculus: Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The first ten terms are 7, 11, 15, 19, 23, 27, 31, 35, 39, and 43. The graph is a straight line.

Step 1: Understanding the Sequence Formula

The sequence is given by the formula \(a_n = 4n + 3\). This formula calculates each term of the sequence based on the position \(n\) of the term.

Step 2: Calculating the First Ten Terms

To find the first ten terms of the sequence, substitute \(n\) with values from 1 to 10 in the formula:- When \(n = 1\), \(a_1 = 4 \times 1 + 3 = 7\)- When \(n = 2\), \(a_2 = 4 \times 2 + 3 = 11\)- When \(n = 3\), \(a_3 = 4 \times 3 + 3 = 15\)- When \(n = 4\), \(a_4 = 4 \times 4 + 3 = 19\)- When \(n = 5\), \(a_5 = 4 \times 5 + 3 = 23\)- When \(n = 6\), \(a_6 = 4 \times 6 + 3 = 27\)- When \(n = 7\), \(a_7 = 4 \times 7 + 3 = 31\)- When \(n = 8\), \(a_8 = 4 \times 8 + 3 = 35\)- When \(n = 9\), \(a_9 = 4 \times 9 + 3 = 39\)- When \(n = 10\), \(a_{10} = 4 \times 10 + 3 = 43\)Thus, the first ten terms of the sequence are 7, 11, 15, 19, 23, 27, 31, 35, 39, 43.

Step 3: Graphing the Sequence

Using a graphing calculator, plot the points \((n, a_n)\) for \(n = 1\) to \(n = 10\). This would involve entering each term calculated in Step 2 as a set of coordinates: (1, 7), (2, 11), (3, 15), (4, 19), (5, 23), (6, 27), (7, 31), (8, 35), (9, 39), and (10, 43). The graph should show these points forming a straight line, as the sequence is linear with a constant difference of 4 between consecutive terms.

Key Concepts

Linear Sequences

Linear sequences are a type of arithmetic sequence where the difference between each consecutive term is constant. In the sequence formula \(a_n = 4n + 3\), the term \(4n\) dictates the linear nature because it grows uniformly as \(n\) increases.
To identify a linear sequence, look for patterns:

• Each term increases (or decreases) by the same amount. Here, the terms increase by 4.

The formula \(a_n = 4n + 3\) tells us that each term is four times the position number \(n\), plus an additional 3. This constant addition of 3 is the offset from zero due to the sequence's specific start location at 7, not 0. Linear sequences are often described as having a **common difference**, which in this case is 4.
Understanding linear sequences is crucial in mathematics because they are simple yet foundational, leading to deeper mathematical concepts like linear functions and equations.

Graphing Sequences

Graphing sequences involves plotting the terms of a sequence on a coordinate plane. For our exercise, you'd plot each term of the sequence \((n, a_n)\) where \(n\) is the position number and \(a_n\) is the term value.
When you plot a linear sequence like \(a_n = 4n + 3\):

• The x-axis represents the term position \(n\).
• The y-axis shows the term value \(a_n\).

In our case, when you plot points such as (1, 7), (2, 11), and so on up to (10, 43), you'll observe a straight line. Each step along the x-axis corresponds to a consistent vertical step of 4 units along the y-axis page, thanks to our linear relationship.
The graph of a linear sequence is a valuable tool, visually confirming the sequence's pattern and providing insight into additional terms or trend continuation without further calculations.

Graphing Calculator

Graphing calculators are powerful tools for visualizing sequences and performing complex calculations, making them indispensable for students tackling sequences in math.
To graph a sequence using a graphing calculator:

• Input the sequence formula, such as \(a_n = 4n + 3\), into the calculator.
• Direct the calculator to plot the sequence points from \(n = 1\) to \(n = 10\).

Most graphing calculators have specific modes or functions for entering sequences directly, showing how the terms align on a graph without manual plotting. They efficiently illustrate the linear path of sequences, making patterns such as the constant difference evident at a glance.
Using a graphing calculator not only saves time but enhances understanding, emphasizing the mathematical concepts behind sequence behavior through real-time, dynamic visualization.