Problem 22 - Precalculus: Mathematical 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. $$a_{n}=a_{n-1}-a_{n-2} \quad \text { and } \quad a_{1}=1, a_{2}=3$$

Step-by-Step Solution

Answer

Calculate terms: 1, 3, 2, -1, -3, -2, 1, 3, 2, -1. Plot these points.

Step 1: Understand the Sequence Definition

The sequence is defined recursively, meaning each term is based on previous terms. The formula is \( a_n = a_{n-1} - a_{n-2} \) with given initial conditions \( a_1 = 1 \) and \( a_2 = 3 \).

Step 2: Calculate the First 10 Terms

Start with the initial terms:\( a_1 = 1 \), \( a_2 = 3 \).Now calculate each term using the recursive formula:- \( a_3 = a_2 - a_1 = 3 - 1 = 2 \)- \( a_4 = a_3 - a_2 = 2 - 3 = -1 \)- \( a_5 = a_4 - a_3 = -1 - 2 = -3 \)- \( a_6 = a_5 - a_4 = -3 + 1 = -2 \)- \( a_7 = a_6 - a_5 = -2 + 3 = 1 \)- \( a_8 = a_7 - a_6 = 1 + 2 = 3 \)- \( a_9 = a_8 - a_7 = 3 - 1 = 2 \)- \( a_{10} = a_9 - a_8 = 2 - 3 = -1 \) So, the first 10 terms are 1, 3, 2, -1, -3, -2, 1, 3, 2, -1.

Step 3: Plot the Sequence

Using a graphing calculator, input the coordinates for each term as \((1,1), (2,3), (3,2), (4,-1), (5,-3), (6,-2), (7,1), (8,3), (9,2), (10,-1)\). Plot these points on a coordinate plane to visually represent the sequence.

Key Concepts

Recursive Sequences

Recursive sequences are a fascinating topic in mathematics. They are sequences where each term is defined using one or more of the preceding terms. In the case of the sequence we're examining, the formula is given by\[a_n = a_{n-1} - a_{n-2}\]with the initial terms provided as \(a_1 = 1\) and \(a_2 = 3\). What makes recursive sequences interesting is their reliance on starting values. The sequence unfolds its long list of numbers based on these initial conditions and its recursive relationship.
This means that any change in the starting values or the formula can lead to a different sequence altogether. Effectively, recursive sequences can demonstrate complex behavior even if the rules are relatively simple. This process is essential for developing problem-solving skills, as it involves identifying patterns and recalculating constantly.
To get recursive sequences down, one must compute each term step-by-step until reaching the desired point within the sequence, in this case, the first 10 terms. Observing the pattern, you can make predictions or analyze the behavior of the sequence over a larger scope.

Graphing Calculator

Graphing calculators are powerful tools for visualizing mathematical concepts, including recursive sequences. They allow you to input the recursive formula and plot the sequence's terms effectively. This visualization helps to better comprehend complex sequences by seeing the relationship between each term plotted on a graph.
When dealing with a sequence like the one in the problem, you can enter the initial conditions \(a_1 = 1\) and \(a_2 = 3\) into the calculator. Then, input the recursive relation to calculate the subsequent terms.A graphing calculator typically offers:

• Capability to handle various mathematical expressions.
• Functionality to plot points as coordinates, making it easier to observe patterns or rotations in sequences.
• Graphical interface that provides clarity beyond numerical data alone.

Graphing calculators can make complicated math problems more manageable by turning numbers into images thus providing a deeper understanding of the relationships and trends within the data.

Sequence Plotting

Sequence plotting is the practice of depicting terms of a sequence on a graph to study their behavior and observe patterns more clearly. For our recursive sequence, each term is paired with its position in the sequence as \((n, a_n)\).This process creates a visual representation where each point is plotted on a coordinate plane.
In the context of the problem, you plot from \((1, 1)\) to \((10, -1)\). Each plotted point represents the term number and its value, making it easier to track changes or cycles as the sequence advances.
Sequence plotting brings several advantages:

• It simplifies the detection of periodic behavior or trends in the data.
• Provides an easier understanding of sequences that may have periodic or oscillating paths.
• Makes transitions and recursive relations within the sequence more visible and manageable.

By examining the graph, insights into whether the sequence diverges, converges, or repeats can be more easily estimated, offering a visual approach to comprehending the structure and nature of recursive sequences.