Problem 29 - Algebra for College Students

Question

Write a formula for the general term of each infinite sequence. \(1,-1,1,-1, \ldots\)

Step-by-Step Solution

Answer

The general term is a_n = (-1)^{n-1}.

Step 1: Identify the pattern

Observe the given sequence: 1, -1, 1, -1,.... Notice that the terms alternate between 1 and -1.

Step 2: Determine the mathematical expression

Since the terms alternate between 1 and -1, we need to find a way to represent this alternation. One common way to represent alternating signs is by using (-1)^n. For even n, (-1)^n = 1, and for odd n, (-1)^n = -1.

Step 3: Generalize the formula

In the given sequence, the 1st term is 1 ((-1)^{1-1} = (-1)^0 = 1), the 2nd term is -1 ((-1)^{2-1} = (-1)^1 = -1), and so on. The general term can be expressed as: a_n = (-1)^{n-1}

Key Concepts

general term formula

The general term formula helps us find any term in an infinite sequence without having to list all the terms. For the given sequence, 1, -1, 1, -1, ... , recognizing the pattern is crucial. Each term alternates between 1 and -1. Using this information, we can create a formula. The formula involves the term \((-1)^{n-1}\). This is because for any given term, n:

• When n is odd, \((-1)^{n-1}\) equals 1 (e.g., n = 1, 3, 5, ...).
• When n is even, \((-1)^{n-1}\) equals -1 (e.g., n = 2, 4, 6, ...).

The general term formula for this sequence is: \[a_n = (-1)^{n-1}\]. By using this formula, we can determine any term in the sequence with ease, making it highly practical for infinite sequences.

alternating sequence

An alternating sequence is a sequence where the terms switch back and forth between two values. In this case, the values are 1 and -1. Here are some key points:

• Alternating sequences often follow a regular pattern.
• They can be identified by the consistent change in signs of consecutive terms (positive, negative, positive, negative, etc.).
• The general form of an alternating sequence is often represented using a power of -1, because \((-1)^n\) cycles through 1 and -1.

Understanding alternating sequences is important because it helps in recognizing patterns and forming formulas quickly. For our sequence, the repeating pattern of 1 and -1 is effectively captured by the general term \((-1)^{n-1}\). This concise representation simplifies calculations and reveals deeper properties of the sequence.

mathematical expression

A mathematical expression is a combination of numbers, operators, and variables that represents a value or a relationship. In sequences, expressions are used to describe the value of each term in relation to its position (n). Here's how mathematical expressions are crafted:

• Identify the pattern or rule that governs the sequence.
• Translate this pattern into mathematical symbols.
• Ensure that the expression correctly reflects the sequence for all terms.

In the provided sequence 1, -1, 1, -1, ..., we used the expression \((-1)^{n-1}\) to alternate between 1 and -1. This expression encapsulates the core behavior of the sequence, turning our verbal observations into a precise and usable formula. Expressions like these are essential tools in mathematics, enabling us to generalize, predict, and understand sequences and patterns efficiently.