Problem 14 - Thomas Calculus

Question

In Exercises \(13-26,\) find a formula for the \(n\) th term of the sequence. The sequence \(-1,1,-1,1,-1, \ldots\)

Step-by-Step Solution

Answer

The formula for the nth term is (-1)^n.

Step 1: Identify the pattern of the sequence

Observe that the sequence is alternating between -1 and 1. The pattern clearly alternates every subsequent term (-1, 1, -1, 1, etc.).

Step 2: Determine the periodicity

The sequence has a period of 2, meaning every two terms, the sequence repeats.

Step 3: Use powers to represent alternating signs

The sequence alternates between -1 and 1. Consider the expression (-1)^n, which will alternate between -1 and 1 as n changes. For instance, when n is odd, (-1)^n = -1; when n is even, (-1)^n = 1.

Step 4: Define the formula for the nth term

The sequence can be represented as (-1)^n. For n = 1, 2, 3 ... the terms generated are -1, 1, -1, 1, ..., which match the sequence provided by the problem.

Key Concepts

Alternating Sequences

An alternating sequence is a type of sequence where the terms switch between different values based on a fixed pattern. In this exercise, the sequence alternates between -1 and 1. Alternating sequences are important because they help us understand cyclic behaviors and repetitive patterns in mathematics. In this example, the sequence flips from -1 to 1 with every new term.

• These sequences can often be expressed using a formula that involves powers of a negative number, like \((-1)^n\). This is a clever way to capture the repeating nature of alternating terms.
• We see these sequences in various mathematical scenarios, such as in the calculation of Fourier series or in different summation techniques.

Understanding alternating sequences often involves recognizing the core pattern and then finding a mathematical way to express it succinctly.

nth Term

The "nth term" of a sequence refers to a formula that allows us to calculate any term in the sequence based on its position, 'n', in the list of terms. This is extremely useful, as it allows us to find distant terms without calculating all preceding terms. In our example of the sequence \(-1, 1, -1, 1, \ldots\), the nth term is given by the formula \((-1)^n\).

• This formula tells us directly what the value of the terms will be for any given 'n'.
• It provides a quick insight into the behavior of the sequence for large 'n'.
• For example, if you need the 100th term, you just calculate \((-1)^{100}\), which equals 1 because 100 is even.

Knowing the nth term functions as a powerful tool to decode the sequence and predict further terms.

Mathematical Patterns

Mathematical patterns are everywhere in sequences and are key to unlocking solutions to complex problems. Identifying a pattern within a sequence can help to establish a formula for the nth term. Often, patterns are the first step in understanding sequences deeply. Given this exercise, the pattern \(-1, 1, -1, 1, \ldots\) is very straightforward; it simply repeats itself every two terms.

• Recognizing patterns can transform seemingly random sequences into predictable sequences governed by mathematical laws.
• Finding patterns relies on looking for repetitions, symmetry, or predictable changes in the sequence.
• Exploring patterns also enhances problem-solving skills and encourages logical thought processes.

By practicing the identification of mathematical patterns, students develop a keen eye for discovering the structure behind the numbers.