Problem 24 - 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. $$a_{n}=a_{n-1}-a_{n-2} \quad \text { and } \quad a_{1}=1, a_{2}=3$$

Step-by-Step Solution

Answer

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

Step 1: Understand the Sequence Formula

The sequence is defined as \(a_{n}=a_{n-1}-a_{n-2}\), with initial values \(a_1=1\) and \(a_2=3\). This means each new term is the difference of the previous two terms.

Step 2: Calculate the First 10 Terms

Using the given sequence formula and initial values, calculate the first 10 terms.\- \(a_1 = 1\)- \(a_2 = 3\)- \(a_3 = a_2 - a_1 = 3 - 1 = 2\)- \(a_4 = a_3 - a_2 = 2 - 3 = -1\)- \(a_5 = a_4 - a_3 = -1 - 2 = -3\)- \(a_6 = a_5 - a_4 = -3 + 1 = -2\)- \(a_7 = a_6 - a_5 = -2 + 3 = 1\)- \(a_8 = a_7 - a_6 = 1 + 2 = 3\)- \(a_9 = a_8 - a_7 = 3 - 1 = 2\)- \(a_{10} = a_9 - a_8 = 2 - 3 = -1\).

Step 3: Graph the Sequence

Using the graphing calculator, plot the points \((n, a_n)\) for \(n=1\) to \(n=10\). The points to plot are: \((1, 1), (2, 3), (3, 2), (4, -1), (5, -3), (6, -2), (7, 1), (8, 3), (9, 2), (10, -1)\). Observe the pattern, noting the repeated cycle in values.

Key Concepts

Graphing Calculator

A graphing calculator is an incredibly helpful tool for sequences and series, especially when visualizing complex relationships. It allows you to plot individual terms of a sequence on a graph, which can help reveal patterns or cycles. When graphing the sequence described in the exercise, you'll input each term as a point in the form \(n, a_n\), where \(n\) is the term number and \(a_n\) is the term's value.
To get started, enter the sequence formula into your calculator. You'll first set the initial conditions as given: \(a_1 = 1\) and \(a_2 = 3\). Then, use the recursive formula \(a_n = a_{n-1} - a_{n-2}\) to generate the subsequent terms sequentially. After you've calculated these terms, input them one by one to create the corresponding points on your graph.
This visual representation can make it much easier to identify if there's a periodic cycle or any irregularities among the numbers. Plus, seeing the results plotted can deepen your understanding of the underlying behavior of the sequence.

Recursive Sequence

A recursive sequence is one where each term is calculated based on the previous terms in the sequence. In this exercise, the sequence is defined recursively by the formula \(a_n = a_{n-1} - a_{n-2}\). This means each term depends on the last two terms, a common characteristic of recursive sequences.
The beauty of recursive sequences is that they operate like a set of instructions; by following them step by step, you can compute any term in the sequence without knowing distance in advance. Starting with the initial values \(a_1 = 1\) and \(a_2 = 3\), the recursive formula helps you determine all subsequent terms. For every new term, subtract the term before last from the last term.
This specific sequence, for instance, creates a cycle that repeats over time, which is often the case in recursively defined sequences. Understanding the iterated process is key to mastering them, and knowing that these sequences typically need initial conditions to start is crucial.

Initial Conditions

Initial conditions are foundational in recursive sequences. They are the starting values that allow the sequence to begin. Without them, the recursive formula wouldn't have a base to start calculating further terms.
For our sequence, the initial conditions are given as \(a_1 = 1\) and \(a_2 = 3\). These values are essential because the sequence relies on them to compute subsequent terms using the given recursive formula \(a_n = a_{n-1} - a_{n-2}\).
Think of initial conditions as the spark that ignites the sequence. They guarantee that each newly calculated term follows logically from the terms before it. Properly setting these conditions ensures that the entire sequence unfolds as intended, with each term building on the ones that came before. Understanding the role of these initial values can provide clearer insight into how recursive sequences evolve over time, and are a critical concept in sequence and series problems.