Problem 32 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the table feature of a graphing utility to find the first 10 terms of the sequences. (Assume \(n\) begins with 1.) $$a_{n}=(-1)^{n+1}+8$$

Step-by-Step Solution

Answer

The first 10 terms of the sequence \(a_{n}=(-1)^{n+1}+8\) are: 9, 7, 9, 7, 9, 7, 9, 7, 9, 7.

Step 1: Understand the problem and sequence formula

The formula provided in the exercise is \(a_{n}=(-1)^{n+1}+8\). To fully comprehend it, it's important to know that \((-1)^n\) will alternate between -1 and 1, depending on whether \(n\) is odd or even. When -1 is raised to an even power, the result is 1. When -1 is raised to an odd power, the result is -1. Thus, the \((-1)^{n+1}\) part of the formula will toggle between 1 (if \(n+1\) is even) and -1 (if \(n+1\) is odd). This will create an alternating pattern in the sequence.

Step 2: Calculate the sequence terms

Now, plug the first ten natural numbers (1, 2, 3, ..., 10) into the formula to find the corresponding terms of the sequence. For \(n=1\), the term is \((-1)^{1+1}+8 = 1+8 = 9\). For \(n=2\), the term is \((-1)^{2+1}+8 = -1+8 = 7\). If you continue this process, you see that the sequence alternates between 9 and 7 for the first ten terms.

Step 3: Summary of the results

The first 10 terms of the sequence are: 9, 7, 9, 7, 9, 7, 9, 7, 9, 7

Key Concepts

Graphing Utility

Graphing utilities are digital tools or software used to visualize mathematical functions and sequences. They are particularly helpful in identifying patterns and relationships in mathematical data.
These tools allow you to input an equation or sequence formula, and then automatically generate a graph or table of values.
When dealing with sequences, such as the one given in your exercise, a graphing utility can swiftly compute and display the first several terms, making it easier to see patterns.
A step-by-step approach involves entering the sequence formula into the utility. In this case, the formula is \(a_{n}=(-1)^{n+1}+8\).

• First, input the formula into your graphing utility, specifying that \(n\) starts at 1.
• Then, use the table view to print out the values for the first ten terms.

This approach minimizes human error and enhances your understanding through visual presentation.

Sequence Formula

A sequence formula defines the rule to calculate each term in a sequence based on its position, \(n\), in the sequence.
The sequence formula provided in this exercise is \(a_{n}=(-1)^{n+1}+8\).
This kind of formula is helpful as it directly gives the computation needed to find any term without having to list all previous terms.

• The part \((-1)^{n+1}\) indicates that the signs alternate. This means each term's sign is dependent on whether \(n+1\) is even or odd.
• The constant \(+8\) shifts all calculated values of \((-1)^{n+1}\) upward by 8, altering the sequence range.

Calculating terms using this formula is systematic: for each \(n\), determine if \(n+1\) is even or odd, compute \((-1)^{n+1}\), and add 8.

Alternating Sequences

Alternating sequences change from positive to negative (or vice versa) as the sequence progresses.
The given sequence formula, \(a_{n}=(-1)^{n+1}+8\), results in such a pattern.
Understanding how these sequences work is crucial; they often relate to functions like \((-1)^n\).
In this exercise, the sequence alternates between two values, 9 and 7, because of the alternating sign component.

• When \(n+1\) is even, \((-1)^{n+1}\) equals 1, resulting in \(a_{n}=1+8=9\).
• When \(n+1\) is odd, \((-1)^{n+1}\) equals -1, giving \(a_{n}=-1+8=7\).

This alternation results in a pattern that repeats itself every two terms. This feature is what defines alternating sequences.