Problem 22 - 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. \(a_{n}=a_{n-1}-a_{n-2} \quad\) and \(\quad a_{1}=1, a_{2}=3\)

Step-by-Step Solution

Answer

First 10 terms: 1, 3, 2, -1, -3, -2, 1, 3, 2, -1. Graph them as points on a graphing calculator.

Step 1: Understanding the Recursive Formula

We are given a recursive sequence where each term is defined as the difference of the two preceding terms. Specifically, for this sequence, we have \(a_n = a_{n-1} - a_{n-2}\) with initial terms \(a_1 = 1\) and \(a_2 = 3\). Our task is to find the first 10 terms.

Step 2: Calculate Term 3

To find \(a_3\), we apply the formula using \(a_1\) and \(a_2\): \(a_3 = a_2 - a_1 = 3 - 1 = 2\). This gives us the third term, \(a_3 = 2\).

Step 3: Calculate Term 4

Using \(a_2\) and \(a_3\), calculate \(a_4\): \(a_4 = a_3 - a_2 = 2 - 3 = -1\). Thus, \(a_4 = -1\).

Step 4: Calculate Term 5

Use \(a_3\) and \(a_4\) to find \(a_5\): \(a_5 = a_4 - a_3 = -1 - 2 = -3\). Therefore, \(a_5 = -3\).

Step 5: Calculate Term 6

Apply the formula using \(a_4\) and \(a_5\): \(a_6 = a_5 - a_4 = -3 - (-1) = -3 + 1 = -2\). Thus, \(a_6 = -2\).

Step 6: Calculate Term 7

Derive \(a_7\) using \(a_5\) and \(a_6\): \(a_7 = a_6 - a_5 = -2 - (-3) = -2 + 3 = 1\). Hence, \(a_7 = 1\).

Step 7: Calculate Term 8

Find \(a_8\) using \(a_6\) and \(a_7\): \(a_8 = a_7 - a_6 = 1 - (-2) = 1 + 2 = 3\). So, \(a_8 = 3\).

Step 8: Calculate Term 9

Use \(a_7\) and \(a_8\) to determine \(a_9\): \(a_9 = a_8 - a_7 = 3 - 1 = 2\). Thus, \(a_9 = 2\).

Step 9: Calculate Term 10

Finally, calculate \(a_{10}\) using \(a_8\) and \(a_9\): \(a_{10} = a_9 - a_8 = 2 - 3 = -1\). Therefore, \(a_{10} = -1\).

Step 10: List the First 10 Terms

The first 10 terms of the sequence are: 1, 3, 2, -1, -3, -2, 1, 3, 2, -1.

Step 11: Graph the Sequence

Use a graphing calculator to plot these 10 terms as points, where the x-axis represents the term number (from 1 to 10) and the y-axis represents the value of each term. This visualizes the alternating pattern evident in the sequence values.

Key Concepts

Recursive Sequences

Recursive sequences fundamentally rely on the idea of defining terms through previous terms. In essence, a recursive formula like the one given helps generate each new term based on the calculations involving preceding terms. For instance, with our sequence defined by the equation \(a_n = a_{n-1} - a_{n-2}\), each term is simply the result of subtracting the second preceding term from the first.Understanding recursive sequences is crucial because they can model various real-life phenomena and present a step-by-step way of handling complexities by building upon simpler, known parts. In our sequence, starting values are essential as they set the stage for all subsequent calculations, making \(a_1 = 1\) and \(a_2 = 3\) foundational for constructing further terms.

Graphing Calculator

A graphing calculator is an incredibly useful tool when dealing with sequences, especially recursive ones. It allows you to input the sequence's formula and its initial conditions, and then automatically computes the subsequent terms.To use a graphing calculator effectively:

• First, ensure the calculator is capable of handling sequences and has the recursive calculation feature.
• Enter the initial terms of the sequence, such as \(a_1 = 1\) and \(a_2 = 3\).
• Input the recursive formula \(a_n = a_{n-1} - a_{n-2}\).
• Use the graphing feature to plot these calculated terms, enabling you to see their distribution and any patterns visually.

The graphic representation of sequences can enhance understanding, especially in identifying the behavior and trends that are less obvious when viewing raw data.

Term Calculation

Calculating terms in a recursive sequence can seem daunting at first, but with a straightforward approach, it becomes more manageable. Begin by determining the value of each succeeding term using the formula provided, ensuring each term is calculated using the prior two terms acknowledged through logical steps.As you proceed through calculations:

• Keep a list of each term calculated for easy reference — from \(a_1 = 1\), \(a_2 = 3\), continuing through each iteration.
• Always substitute the values from calculations of previous terms into the formula as you proceed.
• Each step builds upon the last, like solving a puzzle one piece at a time, which fosters deeper understanding and accuracy with practice.

By practicing term calculation in recursive sequences, one develops the skill to anticipate patterns and identify recurring cycles or alternate trends.

Sequence Visualization

Visualizing a sequence is perhaps one of the most potent methods for understanding its behavior and characteristics. A graph can turn abstract numbers into a clearer pattern, making trends readily apparent. With sequence visualization:

• The x-axis represents the term number sequentially (\(n = 1\), \(n = 2\), etc.), while the y-axis indicates each term's calculated value.
• Plotting each point connects data points in a graph that not only shows where the sequence starts but also how it develops over time.
• Patterns like oscillation, convergence, or divergence can often be easily deduced visually, contributing to intuitive insights into the sequence’s behavior.

The process of visualizing the sequence can reinforce understanding, providing a tangible sense of the mathematical relationships within the sequence.