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Chapter 13: Problem 30

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

Short Answer

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

Step by step solution

01

- Identify the Pattern

Observe how the signs of the terms alternate between -1 and 1. This alternating sign suggests the presence of \((-1)^n\) in the formula.
02

- Determine the Starting Point

Specify whether the first term (which is -1) corresponds to n = 0 or n = 1. Here, the sequence starts from term 1 as \((-1)^1 = -1\).
03

- Write the General Term Formula

Based on the observation, the general term of the sequence can be written as \(a_n = (-1)^n\), where \(n\) starts from 1.

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Key Concepts

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

General Term Formula
In mathematics, a general term formula is a formula that defines the nth term of a sequence. For the given sequence (-1, 1, -1, 1, \textellipsis), determining the general term formula allows us to find any term of the sequence without listing all preceding terms.

To identify the general term formula for an infinite sequence, follow these steps:
  • Pattern Identification: Observe the repetitive changes in the sequence, often involving alternation, increments, or other regular transformations.
  • First Term Identification: Determine whether the first term begins when n = 0 or n = 1.
  • Formula Construction: Combine these observations to create a function that generates every term in the sequence.
For this sequence, the alternation between -1 and 1 suggests that the general term can be given as \( a_n = (-1)^n \). This formula efficiently produces the sequence's terms, starting from n = 1.
Alternating Sequence
An alternating sequence is a sequence of numbers where terms alternate in sign. In simpler terms, positive and negative numbers appear in turns.

In the given sequence (-1, 1, -1, 1, \textellipsis), we can see a clear pattern where each term flips in sign compared to the preceding one. This changing of signs is characteristic of an alternating sequence.

In mathematical notation, this can be represented as \( (-1)^n \), where n is the term number. When n is odd, \( (-1)^n \) results in -1, and when n is even, \( (-1)^n \) results in 1. By understanding this concept, it becomes easier to recognize and formulate alternating sequences.
Pattern Recognition
Recognizing patterns is crucial for identifying how a sequence behaves. Pattern recognition helps break down seemingly complex sequences into manageable parts that follow predictable rules.

Let's take the provided sequence (-1, 1, -1, 1, \textellipsis) as an example. Some key steps in pattern recognition include:
  • Observation: Carefully observe the sequence, noting any repeating elements or changes.
  • Analysis: Look for consistent changes between the terms. In this case, the alternation of signs.
  • Mathematical Expression: Translate the observed pattern into a mathematical form, such as \( (-1)^n \).
Using pattern recognition, we can see that each term switches from -1 to 1, which tells us that the formula includes \( (-1)^n \) to account for this alternation. This clarity allows us to create the general term formula and understand the sequence structure deeply.

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Most popular questions from this chapter

Write the first four terms of the infinite sequence whose nth term is given. \(c_{n}=(-1)^{n}(n-2)^{2}\)

A football is on the 8-yard line, and five penalties in a row are given that move the ball half the distance to the (closest) goal. Write a sequence of five terms that specify the location of the ball after each penalty.

The first two terms of the Fibonacci sequence are 0 and \(1 .\) Every term thereafter is the sum of the two previous terms. So the third term is \(1,\) the fourth term is 2 the fifth term is \(3,\) and the sixth term is \(5 .\) So the first 6 terms of the Fibonacci sequence are \(0,1,1,2,3,5\). a) Write the first 10 terms of the Fibonacci sequence. b) Find an application of the Fibonacci sequence by doing a search on the Internet.

Consider the sequence whose \(n\) th term is \(a_{n}=(0.999)^{n}\). a) Calculate \(a_{100}, a_{1000},\) and \(a_{10,000}\). b) What happens to \(a_{n}\) as \(n\) gets larger and larger?

Write a formula for the general term of each infinite sequence. \(0,1,4,9,16, \dots\)
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