Problem 13 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

The nth term of the sequence is given by \(a_n = (-1)^{n+1}\).

Step 1: Identify the Sequence Pattern

Looking at the sequence, we observe that it alternates between the numbers 1 and -1. Specifically, all odd-position terms (1st, 3rd, 5th, etc.) are equal to 1, and all even-position terms (2nd, 4th, 6th, etc.) are equal to -1.

Step 2: Analyze the Alternating Pattern

Since the sequence alternates every step, we consider using a mathematical method to represent alternating signs. The term \((-1)^{n+1}\) results in alternating 1 and -1. For odd \(n\), \((-1)^{n+1} = 1\), and for even \(n\), \((-1)^{n+1} = -1\).

Step 3: Express the nth Term Formula

From the identified pattern, the nth term can be expressed as \( a_n = (-1)^{n+1} \). This formula will produce 1 for odd \(n\) and -1 for even \(n\).

Key Concepts

Alternating Series

An alternating series is a sequence of numbers where the terms switch signs consistently. In simpler terms, the sequence has a pattern where positive terms are followed by negative terms or vice versa. This pattern of alternating signs is key for understanding certain mathematical sequences.

In the given exercise, the sequence alternates between 1 and -1, creating a consistent pattern. Alternating series are often used to solve problems involving convergence or establish alternating patterns efficiently. One common approach to handle such series is using powers of (-1). When we see ((-1)^n), it means the sequence of terms will change between positive and negative as we progress through the series.

In this specific example, we express the nth term as ((-1)^{n+1}). This clever use of exponentiation lets us mathematically encode the switching signs.

Nth Term Formula

The nth term formula in mathematics explains the general term for any sequence. It provides a way to find the value of any term without listing all preceding terms. Knowing the nth term expression aids in understanding the sequence fully.

For the sequence (1, -1, 1, -1, 1,...), the nth term formula is necessary for determining the value based on its position in the sequence.

• For odd positions: The term equals 1.
• For even positions: The term equals -1.

To express this sequence mathematically, we use: . This formula means for any odd n, will be 1, while for even n, it will be -1. Thus, the nth term formula succinctly captures the sequence's behavior and pattern.

Mathematical Patterns

Mathematical patterns are recurring sequences commonly found in numbers and structures. Recognizing these patterns can solve complex problems efficiently. They allow mathematicians to predict unknowns and identify solutions without tedious computations.

In our sequence example, each term alternates between 1 and -1, showing a simple but vital pattern. Identifying this pattern allows us to understand the underlying rules that govern changes in the sequence. Patterns aren't just about numbers; they can also reveal symmetry, growth, decay, and more.

• Identifying patterns speeds up problem-solving.
• Creates shortcut routes to mathematical solutions.

Recognizing the simple alternating pattern in this problem helps us derive the nth term using precise and clever mathematical techniques. This understanding forms a fundamental skill in both simple and advanced mathematical problem-solving.