Problem 25 - 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. \(a_{n}=\frac{1}{a_{n-1}} \quad\) and \(\quad a_{1}=2\)

Step-by-Step Solution

Answer

The sequence alternates between 2 and \(\frac{1}{2}\). Plot these points on a graph.

Step 1: Identify the Recursive Formula

The sequence given is defined recursively, with \(a_1 = 2\) and \(a_n = \frac{1}{a_{n-1}}\). This means each term is the reciprocal of the previous term.

Step 2: Calculate the First Ten Terms

To find the first ten terms in the sequence, we calculate each term using the recursive formula, starting with \(a_1 = 2\):- \(a_1 = 2\)- \(a_2 = \frac{1}{a_1} = \frac{1}{2}\)- \(a_3 = \frac{1}{a_2} = 2\)- \(a_4 = \frac{1}{a_3} = \frac{1}{2}\)- \(a_5 = \frac{1}{a_4} = 2\)- \(a_6 = \frac{1}{a_5} = \frac{1}{2}\)- \(a_7 = \frac{1}{a_6} = 2\)- \(a_8 = \frac{1}{a_7} = \frac{1}{2}\)- \(a_9 = \frac{1}{a_8} = 2\)- \(a_{10} = \frac{1}{a_9} = \frac{1}{2}\)The first ten terms of the sequence are: 2, \(\frac{1}{2}\), 2, \(\frac{1}{2}\), 2, \(\frac{1}{2}\), 2, \(\frac{1}{2}\), 2, \(\frac{1}{2}\).

Step 3: Graph the Sequence Terms

To graph the sequence, plot these terms on a set of axes where the x-axis represents the term number and the y-axis represents the term value. You will plot the points: (1, 2), (2, \(\frac{1}{2}\)), (3, 2), (4, \(\frac{1}{2}\)), (5, 2), (6, \(\frac{1}{2}\)), (7, 2), (8, \(\frac{1}{2}\)), (9, 2), (10, \(\frac{1}{2}\)). The graph will show an alternating pattern between 2 and \(\frac{1}{2}\).

Key Concepts

Recursive Formula

A recursive formula is like a mathematical recipe. It tells you how to find each term in a sequence based on the previous ones. In our exercise, the recursive formula is:\[a_n = \frac{1}{a_{n-1}}\]Starting with the initial term \(a_1 = 2\), the formula shows that to get the next term, you take the reciprocal of the previous term. This means you flip the fraction: if your previous term was 2 (written as \(\frac{2}{1}\)), the next term is its reciprocal: \(\frac{1}{2}\). Then the following term flips back to 2, and so on. This creates a pattern from each term to the next, making it predictable and easy to follow.

Sequence Terms

Sequence terms are essentially the elements or numbers in a sequence that we calculate. Here, each term is either 2 or \(\frac{1}{2}\). Let's visualize them:- Start with \(a_1 = 2\)- Then find \(a_2\) by taking \(\frac{1}{2}\)- Continue this process to get \(a_3 = 2\)- Then \(a_4 = \frac{1}{2}\), and so onFor the first ten terms, you end up with: - 2, \(\frac{1}{2}\), 2, \(\frac{1}{2}\), 2, \(\frac{1}{2}\), 2, \(\frac{1}{2}\), 2, \(\frac{1}{2}\)Notice how these numbers form a repeating cycle. Each term depends directly on the one before it, creating a clear and alternating pattern. Understanding these individual terms helps us see the big picture of how sequences grow.

Graphing Calculator

A graphing calculator is a handy tool for visualizing sequences. By plotting sequence terms on a graph, you can better understand their behavior and patterns. When you enter our sequence terms from the exercise into a graphing calculator, you plot points based on their order:- The x-axis represents the term number, such as 1, 2, 3- The y-axis shows the term's value, like 2 or \(\frac{1}{2}\)Here's what you do:- Input points like (1, 2), (2, \(\frac{1}{2}\)), (3, 2), (4, \(\frac{1}{2}\))- Continue to plot all terms up to (10, \(\frac{1}{2}\))After plotting all the points, the graph will show alternating heights, illustrating the periodic nature of our sequence. This visual helps you spot trends and confirm that calculations are correct.

Alternating Sequence

An alternating sequence is one where terms change back and forth between different values. In our case, the sequence alternates between 2 and \(\frac{1}{2}\). Here's how it works:- Starting at 2, we flip to \(\frac{1}{2}\)- Then flip back to 2, and to \(\frac{1}{2}\) again- This pattern repeats for the entire sequenceAlternating sequences are easy to predict once you identify the pattern. In math, this kind of sequence is useful for modeling situations where values oscillate between extremes. By recognizing the alternating nature, you quickly grasp how the sequence develops and repeats its cycle.