Problem 23 - 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. $$a_{n}=\frac{1}{a_{n-1}} \text { and } a_{1}=2$$

Step-by-Step Solution

Answer

The first 10 terms are: 2, 0.5, 2, 0.5, 2, 0.5, 2, 0.5, 2, 0.5. Graph them as alternating points at y = 2 and y = 0.5.

Step 1: Understand the Sequence Formula

The sequence is defined recursively by the formula \( a_n = \frac{1}{a_{n-1}} \). This means each term in the sequence is the reciprocal of the previous term. The first term of the sequence \( a_1 \) is given as 2.

Step 2: Calculate the First 10 Terms Manually

Using the recursive formula and the initial term, we can calculate the first 10 terms as follows:- \( a_1 = 2 \)- \( a_2 = \frac{1}{2} = 0.5 \)- \( a_3 = \frac{1}{0.5} = 2 \)- \( a_4 = \frac{1}{2} = 0.5 \)- Continue this pattern...Thus, the first 10 terms are: 2, 0.5, 2, 0.5, 2, 0.5, 2, 0.5, 2, 0.5.

Step 3: Graph the Sequence Terms

On a graphing calculator, plot the calculated terms as discrete points. The x-axis represents the term number (from 1 to 10), and the y-axis represents the value of the sequence terms. You should see points alternating between y = 2 and y = 0.5.

Key Concepts

Graphing Calculator

A graphing calculator is an essential tool in visualizing complex mathematical functions and sequences. These calculators are designed to display calculations graphically, making it easier to understand and analyze mathematical concepts. They can plot functions, perform algebraic operations, and represent data visually. In the context of sequences, a graphing calculator allows users to input sequence formulas and quickly visualize the terms. By plotting each term as a point, students can see the behavior and pattern of the sequence over a specified range. For our recurring sequence, a graphing calculator helps by:

• Quickly calculating terms based on the defined formula.
• Providing a visual representation of the sequence.
• Helping students identify patterns and periodicity.

Graphing calculators are thus excellent learning aids for students tackling recursive sequences, giving a better understanding of mathematical structures beyond simple calculations.

Sequence Graphing

Sequence graphing refers to the process of plotting the terms of a sequence to understand its behavior visually. This is particularly useful for recursive sequences, where terms are defined based on previous terms.When graphing a sequence like the one given in the exercise, the plot helps to reveal patterns that might not be immediately obvious from the formula alone. Each term of the sequence is plotted as a discrete point on the graph:

• The x-axis represents the term number.
• The y-axis shows the value of each term.

In our example, you can see how the sequence values alternate between 2 and 0.5 by graphing the first ten terms:\[ a_1, a_2, a_3, ext{…} a_{10} = 2, 0.5, 2, 0.5, ext{…} \]This visual representation allows one to easily grasp the periodic pattern of the sequence. It also helps in predicting future terms and understanding how the sequence behaves as the number of terms increases.

Recurrence Relation

A recurrence relation is a way to define a sequence where each term is determined by the preceding terms. This is in contrast to an explicit formula, which expresses each term independently.For the sequence given in the exercise, the recurrence relation is:\[ a_n = \frac{1}{a_{n-1}} \]This relation shows that each term is the reciprocal of the previous one. Recurrence relations are powerful in mathematical modeling as they simplify complex problems by breaking them down into simpler, iterative steps.

• They help in defining sequences where an explicit formula might be challenging to find.
• They're used extensively in computer algorithms, physics, and other applied sciences.
• Understanding how a recurrence relation works helps in predicting future terms in the sequence effectively.

Given the initial term, the sequence is built iteratively, showcasing a dynamic relationship. This kind of definition is incredibly useful for understanding sequences that exhibit specific repeating patterns.