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Chapter 13: Problem 29

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

Short Answer

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

Step by step solution

01

Identify the pattern

Observe the given sequence: 1, -1, 1, -1,.... Notice that the terms alternate between 1 and -1.
02

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.
03

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}

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

The MSRP for a well-equipped 2007 Ford \(\mathrm{F}-250\) Lariat \(4 \mathrm{WD}\) Super Duty Super Cab was \(\$ 43,568\) (www.edmunds.com). Suppose that the price of this model increases by \(5 \%\) each year. Find the price to the nearest dollar for the 2008 through 2012 models.

Evaluate each expression. $$\frac{9 !}{2 ! 7 !}$$

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Which of the following sequences is not an arithmetic sequence? Explain your answer. a) \(\frac{1}{2}, 1, \frac{3}{2}, \dots\) b) \(\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots\) c) \(5,0,-5, \dots\) d) \(2,3,4, \ldots\)

After reading this secrion, write out the answers to these questions. Use complete sentences. What is Pascal's triangle and how do you make it?
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