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Chapter 11: Problem 16

Decide whether each sequence is finite or infinite. The sequence of pages in a book

Short Answer

Expert verified
The sequence of pages in a book is finite.

Step by step solution

01

Understand the Problem

Determine if the sequence of pages in a book is finite or infinite by considering the nature of the sequence.
02

Identify Characteristics of Finite and Infinite Sequences

A finite sequence has a specific number of terms, whereas an infinite sequence continues indefinitely without end.
03

Analyze the Scenario

A physical book has a specific number of pages, which can be counted. It is physically impossible for a book to have an infinite number of pages.
04

Conclusion

Since a book has a definite and countable number of pages, the sequence of pages in a book is finite.

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Key Concepts

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

finite sequences
Finite sequences are collections of elements that have a specific, countable number of terms. In other words, a finite sequence will always have a beginning and an end. For example, consider the sequence of pages in a book. Each page number follows in a specific order, and you can count how many pages there are. This means the sequence of pages is finite since it has a clear start and end.

Some key characteristics of finite sequences include:
  • A defined first and last term.
  • A countable number of elements.
  • The ability to specify the total number of terms (often denoted as 'n').
Understanding these traits will help you identify finite sequences quickly in various contexts like books, lists of class students, or episodes of a TV series.
infinite sequences
Unlike finite sequences, infinite sequences do not have an end. They continue endlessly and their terms keep going without limit. A common example is the sequence of natural numbers: 1, 2, 3, 4, and so on.

Key characteristics of infinite sequences include:
  • No final term; the sequence goes on forever.
  • It is impossible to count all the elements because they are limitless.
  • Often represented using ellipses (e.g., 1, 2, 3, ...).
Infinite sequences are found in mathematical contexts like number lines and are used to represent ongoing processes or measurements that do not stop.

Since they have no end, infinite sequences are theoretical and help mathematicians understand concepts like limits and infinity.
sequence analysis
Sequence analysis involves examining the properties and behaviors of sequences to understand their nature. For example, consider the sequence of pages in a book analyzed in the solution.

When analyzing sequences:
  • First, determine if the sequence is finite or infinite by checking if it has a countable end.
  • Consider the context of the sequence to understand its characteristics.
  • Identify any patterns or rules that define the sequence.
Through sequence analysis, you can distinguish between finite and infinite sequences and gain deeper insights into their structure and behavior. This skill is important for solving problems in both academic and real-world contexts.

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Most popular questions from this chapter

The table gives the results of a survey of \(219,864\) first-year shudents from the class of 2013 at 297 of the nation's four-year colleges and universities.$$\begin{array}{|l|c|c|c|c}\hline \text { Number of Colleges Applied to } & 1 & 2 \text { or } 3 & 4-6 & 7 \text { or more } \\\\\hline \text { Percent (as a decimal) } & 0.14 & 0.24 & 0.39 & 0.23 \\\\\hline\end{array}$$ Using the percents as probabilities, find the probability of each event for a randomly selected student. The student applied to at least 2 colleges.4 colleges.

Use the sequence feature of a graphing calculator to graph the first ten terms of each sequence as defined. Use the graph to make a conjecture as to whether the sequence converges or diverges. If you think it converges, determine the number to which it converges. $$a_{n}=\left(1+\frac{1}{n}\right)^{n}$$

A baseball team has 20 players. How many 9-player batting orders are possible?

Use the summation feature of a graphing calculator to evaluate each sum. Round to the nearest thousandth. $$\sum_{j=3}^{8} 2(0.4)^{j}$$

Prove statement for positive integers \(n\) and \(r,\) with \(r \leq n\). (Hint: Use the definitions of permutations and combinations.) $$P(n, n)=n !$$
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