Problem 26 - 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. \(a_{n}=a_{n-1}-a_{n-2} \quad\) 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 the terms on a coordinate plane.

Step 1: Understand the Sequence Formula

The given recursive formula for the sequence is \(a_n = a_{n-1} - a_{n-2}\). This means that each term in the sequence is the difference of the two preceding terms. The initial conditions are given as \(a_1 = 1\) and \(a_2 = 3\).

Step 2: Calculate the First Term

According to the sequence's initial condition, \(a_1 = 1\). No calculation is necessary for this step as it is provided.

Step 3: Calculate the Second Term

Similarly, from the information given, \(a_2 = 3\). This is used along with \(a_1\) to find further terms.

Step 4: Calculate the Third Term

Using the formula \(a_n = a_{n-1} - a_{n-2}\), find \(a_3\):\[a_3 = a_2 - a_1 = 3 - 1 = 2\]

Step 5: Calculate the Fourth Term

Find \(a_4\) using the same formula:\[a_4 = a_3 - a_2 = 2 - 3 = -1\]

Step 6: Calculate the Fifth Term

Continue applying the formula to find \(a_5\):\[a_5 = a_4 - a_3 = -1 - 2 = -3\]

Step 7: Calculate the Sixth Term

Find \(a_6\):\[a_6 = a_5 - a_4 = -3 - (-1) = -3 + 1 = -2\]

Step 8: Calculate the Seventh Term

Find \(a_7\):\[a_7 = a_6 - a_5 = -2 - (-3) = -2 + 3 = 1\]

Step 9: Calculate the Eighth Term

Find \(a_8\):\[a_8 = a_7 - a_6 = 1 - (-2) = 1 + 2 = 3\]

Step 10: Calculate the Ninth Term

Find \(a_9\):\[a_9 = a_8 - a_7 = 3 - 1 = 2\]

Step 11: Calculate the Tenth Term

Finally, find \(a_{10}\):\[a_{10} = a_9 - a_8 = 2 - 3 = -1\]

Step 12: List the First Ten Terms

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

Step 13: Graph the First Ten Terms

Using a graphing calculator, plot the sequence with \(n\) on the x-axis and \(a_n\) on the y-axis. Mark the points for each term to create the graph, which displays the oscillating pattern of the sequence.

Key Concepts

Recursive sequences

Recursive sequences are a type of sequence where each term is defined based on one or more of its preceding terms. It requires an initial value or values to "start" the sequence. This is known as an initial condition. In our example:

• The sequence is defined by the formula: \(a_n = a_{n-1} - a_{n-2}\).
• The initial conditions are \(a_1 = 1\) and \(a_2 = 3\).

This means that every term, starting from the third, can be calculated by taking the preceding term and subtracting the term before it. Recursive sequences are immensely useful in various fields such as computer science, mathematics, and even biology, as they often reflect processes that build upon previous stages.
They help to model phenomena where each stage or term depends critically on prior stages.

Graphing calculator

A graphing calculator is an invaluable tool for visualizing and analyzing mathematical sequences and functions. It allows you to:

• Visualize sequences by plotting terms against their position.
• Analyze patterns and behaviors visually, which can sometimes be difficult to discern numerically.

In this exercise, a graphing calculator can be used to plot the first ten terms of the sequence we calculated. By doing this:
You plot each term with the sequence position \(n\) as the x-axis and the term value \(a_n\) as the y-axis.
The plot reveals the oscillating pattern of the sequence - a characteristic that's not immediately obvious from the numerical terms alone. A graph helps to understand the dynamics of the sequence and could lead to deeper insights or hypotheses about its behavior over a larger number of terms.

Sequence terms calculation

Calculating the terms of a recursive sequence involves repeatedly using its formula and initial conditions. Here’s a simple guide to calculate each term:1. Use the initial conditions for the first terms. For instance: - Given \(a_1 = 1\). - Given \(a_2 = 3\).
2. Apply the recursive formula to arrive at each subsequent term. Here: - Calculate \(a_3\) using \(a_2 - a_1\). - Similarly, continue the calculation for \(a_4, a_5,\) and so on.
Each term utilizes two preceding terms. This process iteratively generates the sequence until the desired number of terms is reached. Be careful to perform each arithmetic step correctly, as errors can quickly propagate through a recursive sequence. While manually calculating terms can enhance understanding, using a tool like a graphing calculator can ensure accuracy, especially for complex or lengthy sequences.