Problem 19 - Precalculus Mathematics for Calculus

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, 43. Graph these points to see a straight line.

Step 1: Determine Sequence Formula

The given formula for the sequence is \( a_n = 4n + 3 \), where \( n \) is an integer starting from 1. This formula tells us how to compute each term in the sequence based on its position \( n \).

Step 2: Calculate the First 10 Terms

To find the first 10 terms, substitute values from 1 to 10 for \( n \) in the formula \( a_n = 4n + 3 \). Calculate each one:- For \( n = 1 \): \( a_1 = 4(1) + 3 = 7 \)- For \( n = 2 \): \( a_2 = 4(2) + 3 = 11 \)- For \( n = 3 \): \( a_3 = 4(3) + 3 = 15 \)- For \( n = 4 \): \( a_4 = 4(4) + 3 = 19 \)- For \( n = 5 \): \( a_5 = 4(5) + 3 = 23 \)- For \( n = 6 \): \( a_6 = 4(6) + 3 = 27 \)- For \( n = 7 \): \( a_7 = 4(7) + 3 = 31 \)- For \( n = 8 \): \( a_8 = 4(8) + 3 = 35 \)- For \( n = 9 \): \( a_9 = 4(9) + 3 = 39 \)- For \( n = 10 \): \( a_{10} = 4(10) + 3 = 43 \)

Step 3: Graph the Terms

Using a graphing calculator, input the values for the first 10 terms on a grid. Plot each point \((n, a_n)\) where \( n \) is the position and \( a_n \) is the term's value. The plotted points will be: (1, 7), (2, 11), (3, 15), (4, 19), (5, 23), (6, 27), (7, 31), (8, 35), (9, 39), (10, 43). Connect the points to observe the linear pattern.

Key Concepts

Understanding Sequences

Sequences are an essential concept in mathematics, where they represent ordered lists of numbers that follow specific rules. In this context, we are dealing with a particular type of sequence known as an "arithmetic sequence," where each term is obtained by adding a constant value to the previous term. This constant value is commonly referred to as the "common difference."
For the sequence given by the formula \( a_n = 4n + 3 \), where \( n \) is an integer starting from 1, the common difference is 4. Thus, each term is obtained by adding 4 to the previous one, starting with 7.

• First Term: For \( n = 1 \), \( a_1 = 7 \)
• Second Term: For \( n = 2 \), \( a_2 = 11 \)
• Third Term: For \( n = 3 \), \( a_3 = 15 \)

Understanding such sequences helps us make predictions and analyze mathematical patterns seamlessly. Whether you're examining trends in data or modeling mathematical phenomena, grasping the concept of sequences is vital.

Exploring Linear Functions

Linear functions are one of the fundamental building blocks in mathematics and are defined by an equation of the form \( y = ax + b \). These functions produce straight-line graphs and describe relationships with a constant rate of change.
In our exercise, the sequence \( a_n = 4n + 3 \) also represents a linear function in terms of its sequence position \( n \). Comparing this with the standard linear form, \( a_n \) can be seen as \( y \), \( n \) as \( x \), 4 as the slope (rate of change), and 3 as the y-intercept (base value when \( n = 0 \)).
This means that the sequence grows by 4 for every increase by 1 in \( n \). Understanding linear functions is crucial as they illustrate simple changes and relationships in various contexts, from business growth predictions to everyday phenomena.

Plotting Sequences with a Graphing Calculator

Plotting sequences on a graphing calculator helps visualize the relationship between terms and their positions. This visualization is crucial in understanding how terms increase and the linear nature of the sequence.
To correctly plot the sequence \( a_n = 4n + 3 \), follow these steps:

• Input values for \( n \) from 1 to 10 into the calculator.
• Compute each corresponding term \( a_n \) using the formula.
• Plot each point as \( (n, a_n) \) on the graph, where \( n \) is along the x-axis and \( a_n \) along the y-axis.

By connecting these points, you'll notice that they form a straight line, highlighting the linear relationship. Graphing this on a calculator is not only an excellent way to verify calculated values but also to intuitively grasp the concept of linear patterns in sequences. Using this visually engaging method enhances comprehension and aids in the retention of mathematical principles.