Problem 31 - 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.) $$a_{n}=(-1)^{n}+1$$

Step-by-Step Solution

Answer

The first 10 terms of the sequence are 0, 2, 0, 2, 0, 2, 0, 2, 0, 2.

Step 1: Understand the Sequence Formula

The sequence given is \(a_{n}=(-1)^{n}+1\), where \(n\) is the term number starting from 1. This formula uses exponentiation, and with its application, it generates the sequence.

Step 2: Calculate the First Term

For \(n=1\), substitute \(n\) into the sequence formula: \(a_{1} = (-1)^{1} + 1 = -1 + 1 = 0\). So, the first term is 0.

Step 3: Calculate the Second Term

For \(n=2\), substitute it into the sequence formula: \(a_{2} = (-1)^{2} + 1 = 1 + 1 = 2\). So, the second term is 2.

Step 4: Calculate Terms Until the Tenth

Continue this process for \(n=3\) to \(n=10\) using the sequence formula. Notice that the sequence alternates between 0 and 2.

Step 5: Observe the Sequence Pattern

By looking at the calculated values, you can see that the sequence alternates between 0 when \(n\) is odd, and 2 when \(n\) is even. This is reflected by the \((-1)^n\) part of the formula.

Key Concepts

Sequence Formula

A sequence formula is an expression that defines the terms of a sequence. In the exercise you're exploring, the sequence is given by the formula \(a_{n}=(-1)^{n}+1\). Let's break it down:

• \((-1)^n\) is an alternating component, switching signs depending on whether \(n\) (the term index) is odd or even.
• The "+1" shifts the value of the alternating sequence up by one.

These formulas are vital because they provide a rule to calculate any term in the sequence directly by plugging in the appropriate value of \(n\). This makes analyzing sequences efficient, especially for larger values of \(n\). Understanding the sequence formula helps in identifying patterns and predicting subsequent terms without extensive manual calculations.

Alternating Sequence

An alternating sequence is a type of sequence where the terms alternate between two or more values based on a consistent pattern. In the given sequence, \(a_{n} = (-1)^{n} + 1\), the primary alternating component is \((-1)^n\).
Here’s how it influences the sequence:

• When \(n\) is odd, for example \(n=1\), \((-1)^1 = -1\).
• When \(n\) is even, for example \(n=2\), \((-1)^2 = 1\).

The effect of this alternating sign is reflected in the sequence generated:

• When \(n\) is odd, \(a_n = 0\) (because \(-1+1=0\)).
• When \(n\) is even, \(a_n = 2\) (because \(1+1=2\)).

Understanding alternating sequences can be significant in patterns that repeat themselves periodically, providing insight into the more complex forms of numerical repetition and change.

Graphing Utility

A graphing utility is a tool, either digital like software or a calculator, used to visualize mathematical functions and data. In the context of sequences, graphing utilities are incredibly helpful.
Employing a graphing utility allows you to:

• Plot the terms of a sequence to see visual patterns more readily than with numerical values alone.
• Utilize the table feature to quickly calculate and organize many terms of a sequence.

For this particular exercise, using a graphing utility can help identify the alternating pattern of the sequence: 0, 2, 0, 2, and so on. By seeing these patterns visually, you gain a new perspective on the solution, possibly revealing new insights about behavior across various sequences. Graphing utilities are thus pivotal in exploring the deeper, often more intuitive properties of sequences.

Term Calculation

Term calculation in a sequence means determining the value of each term based on its position (\(n\)) in the series. For sequences defined with a formula like \(a_{n}=(-1)^{n}+1\), term calculation involves substituting different values for \(n\) and simplifying the expression.
Here are steps for term calculation:

• Identify the formula: \(a_{n}=(-1)^{n}+1\).
• Substitute the term number (e.g., \(n=1, 2, 3\), etc.) into the formula.
• Calculate using the operations specified (exponentiation and addition in this case).

For example:

• For \(n=1\), the formula gives \((-1)+1=0\).
• For \(n=2\), it gives \(1+1=2\).

By following these steps, you can find the values of sequence terms systematically, aiding in understanding the overall behavior of the sequence.