Problem 14 - Thomas Calculus

Question

Find a formula for the \(n\)th term of the sequence. $$ -1,1,-1,1,-1, \dots $$

Step-by-Step Solution

Answer

The \(n\)th term is \(a_n = (-1)^n\).

Step 1: Identify the Pattern

Observe the given sequence: -1, 1, -1, 1, -1. Notice that the terms alternate between -1 and 1 consistently. This suggests that the sign of the terms might depend on whether their position in the sequence is odd or even.

Step 2: Determine the Formula

For this sequence, notice that the odd-numbered terms are -1, and the even-numbered terms are 1. A useful function to achieve this alternation is \((-1)^n\), which produces -1 for odd \(n\) and 1 for even \(n\). Therefore, the \(n\)th term of the sequence can be defined as \(a_n = (-1)^n\).

Key Concepts

Alternating Series

An alternating series is a sequence of numbers in which the signs of the terms switch back and forth between positive and negative. This means that with each consecutive term, the sign flips. In our example sequence, the terms alternate between -1 and 1. The magic behind alternating series lies in this regular change of signs, making it unique and easy to identify. Such series are common in mathematics because they often model cyclic behaviors and processes.

Understanding alternating series is crucial as they lay the foundation for more complex series and applications. Other examples of alternating series in mathematics include the alternating harmonic series or alternating geometric sequences, each of which follows their unique patterns based on the idea of switching signs.

Nth Term Formula

The nth term formula is a way to find the value of any term in a sequence without listing down all the preceding terms. For those who are analyzing mathematical patterns, finding the nth term formula is essential as it provides a concise and powerful way to describe the sequence. In the case of our sequence, the formula is given by \(a_n = (-1)^n\).

Here's how it works:

• When \( n \) is odd, \((-1)^n\) gives -1, matching with the sequence's odd positions.
• When \( n \) is even, \((-1)^n\) yields 1, aligning with the sequence's even positions.

This formula takes advantage of powers of -1 to decide the sign of each term, making it a perfect match for our alternating series. Once you master the nth term formula for a sequence, predicting future terms becomes simple and efficient.

Odd and Even Positions

In sequences, recognizing whether a position is odd or even can greatly aid in finding patterns and understanding the sequence's behavior. In our example, the sequence alternates based on the position's parity—that is, whether the position number \( n \) is odd or even.

Why does this matter? Odd and even positions help in developing the strategy to identify term behaviors:

• Odd positions (like the 1st, 3rd, 5th positions) produce a value of -1 in this sequence.
• Even positions (like the 2nd, 4th positions) yield a value of 1.

This concept of parity can also be extended to more complex sequences and series, where alternating behaviors depend on odd and even positional logic. Knowing how to distinguish and utilize positions is a valuable skill in the realm of sequences.