/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 59 In Exercises 47-62, write an exp... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 59

In Exercises 47-62, write an expression for the apparent \(n\)th term of the sequence. (Assume that \( n \) begins with 1.) \( 1, 3, 1, 3, 1, \dots \)

Short Answer

Expert verified
The formula for the nth term of the sequence is \( a_n = 2 + (-1)^n \)

Step by step solution

01

Identifying the Pattern of the Sequence

Take a look at the given sequence. The sequence is 1, 3, 1, 3, 1, and so on. It is clear that the sequence alternates between the numbers 1 and 3. Specifically, the number 1 appears at the odd-numbered terms, and the number 3 appears at the even-numbered terms.
02

Formulate a Mathematical Expression

The nth term of this sequence can be written in the form of a function of \( n \) that models this alternating pattern between 1 and 3. This function is \( a_n = 2 + (-2)^n \).
03

Verify the Proposed Expression

To confirm if a function is correct, replace \( n \) with the term's position number in the sequence for a few positions, such as \( n = 1, 2, 3, 4 \) and see if the expected sequence values are obtained. At \( n = 1 \), \( a_n = 2 + (-2)^1 = 0. At \( n = 2 \), the expression gives \( a_n = 2 + (-2)^2 = 4 \), and this pattern continues on for the subsequent terms. The pattern does not coincide with the original sequence, which means the proposed expression is incorrect.
04

Re-formulate a Correct Mathematical Expression

Let's retry formulating the expression. A correct function who alternates between 1 and 3 could be \( a_n = 2 + (-1)^n \).
05

Verify the Correctness of the New Expression

Check if the new function gives the correct sequence terms. For \( n = 1, 2, 3, ... \) the function \( a_n = 2 + (-1)^n \) gives: 1, 3, 1, 3, and so on. This matches the original sequence, so the correct function for the nth term of this sequence is indeed \( a_n = 2 + (-1)^n \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Alternating Sequences
An alternating sequence is a type of sequence in which the terms switch between specific values or patterns regularly. For example, in the sequence given in the exercise, the terms alternate between the numbers 1 and 3. This type of sequence can be visually identified through:
  • The repetition of different numbers or patterns in a regular way.
  • An observation where the sequence clearly changes its value based on whether the term鈥檚 position is even or odd.
Recognizing an alternating sequence is an essential skill for understanding how sequences behave and how to express them mathematically. Alternating sequences often require creative thinking for formulating expressions that capture these changes.
Sequence Pattern
Identifying a sequence pattern involves looking for a recurring theme or rule that defines how each term relates to its position in the sequence. In the given sequence, the pattern is straightforward:
  • The number 1 appears at odd-numbered positions: 1st, 3rd, 5th, etc.
  • The number 3 appears at even-numbered positions: 2nd, 4th, 6th, etc.
By recognizing this pattern, one can easily predict future terms or construct a formula that represents it. Understanding such patterns allows us to express the sequence mathematically, further aiding in mathematical calculations and problem-solving.
Sequence Expression
A sequence expression is a formula that describes each term鈥檚 value in relation to its position number, known as \( n \). For alternating sequences, expressions often include factors or functions that switch values based on \( n \). In the corrected solution for the exercise, the expression \( a_n = 2 + (-1)^n \) was used.
  • The term \((-1)^n\) alternates between -1 and 1 depending on whether \( n \) is odd or even.
  • Adding 2 shifts the whole sequence to alternate between 1 and 3.
Creating the right expression involves verifying its correctness by substituting different \( n \) values to see if it produces the expected sequence. Once confirmed, this expression becomes an efficient way to describe the sequence mathematically, allowing easy calculation of any term without listing all previous terms.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In Exercises 1 - 7, fill in the blanks. If \( P\left(E\right) = 0 \), then \( E \) is an ______ event, and if \( P\left(E\right) = 1 \), then \( E \) is a _______ event.

In Exercises 9 - 14, determine the sample space for the experiment. A coin and a six-sided die are tossed.

In Exercises 15 - 20, find the probability for the experiment of tossing a coin three times. Use the sample space \( S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} \). The probability of getting at least two heads

Three points that are not collinear determine three lines. How many lines are determinedby nine points, no three of which are collinear?

Write a short paragraph defining the following. (a) Sample space of an experiment (b) Event (c) The probability of an event \( E \) in a sample space \( S \) (d) The probability of the complement of \( E \)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.