/*! 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 37 Write the terms \(a_{1}, a_{2}, ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 37

Write the terms \(a_{1}, a_{2}, a_{3},\) and \(a_{4}\) of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. $$a_{n+1}=1+\frac{a_{n}}{2} ; a_{0}=2$$

Short Answer

Expert verified
Does the sequence converge, and if so, what is its limit? Answer: The first four terms of the sequence are \(a_1 = 2\), \(a_2 = 2\), \(a_3 = 2\), and \(a_4 = 2\). The sequence converges, and its limit is 2.

Step by step solution

01

Find the first term a_1

Using the given recurrence relation, we can find \(a_1\) by plugging \(a_0\) into the formula: $$a_{1} = 1 + \frac{a_{0}}{2} = 1 + \frac{2}{2} = 1 + 1 = 2.$$
02

Find the second term a_2

Similarly, we can find \(a_2\) by plugging \(a_1\) into the formula: $$a_{2} = 1 + \frac{a_{1}}{2} = 1 + \frac{2}{2} = 1 + 1 = 2.$$
03

Find the third term a_3

Continue the process to find \(a_3\): $$a_{3} = 1 + \frac{a_{2}}{2} = 1 + \frac{2}{2} = 1 + 1 = 2.$$
04

Find the fourth term a_4

Lastly, find \(a_4\) using the same method: $$a_{4} = 1 + \frac{a_{3}}{2} = 1 + \frac{2}{2} = 1 + 1 = 2.$$
05

Analyze the sequence for convergence

After calculating the first four terms of the sequence, we can see that they are all equal to 2. This suggests that the sequence converges to a constant value. Since all the terms we have found are equal to 2, we can make a conjecture that the sequence converges to a limit of 2.

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!

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

A glimpse ahead to power series Use the Ratio Test to determine the values of \(x \geq 0\) for which each series converges. $$\sum_{k=1}^{\infty} \frac{x^{k}}{2^{k}}$$

Consider the following situations that generate a sequence. a. Write out the first five terms of the sequence. b. Find an explicit formula for the terms of the sequence. c. Find a recurrence relation that generates the sequence. d. Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist. When a biologist begins a study, a colony of prairie dogs has a population of \(250 .\) Regular measurements reveal that each month the prairie dog population increases by \(3 \%\) Let \(p_{n}\) be the population (rounded to whole numbers) at the end of the \(n\) th month, where the initial population is \(p_{0}=250\).

Use Exercise 89 to determine how many terms of each series are needed so that the partial sum is within \(10^{-6}\) of the value of the series (that is, to ensure \(R_{n}<10^{-6}\) ). a. \(\sum_{k=0}^{\infty} 0.6^{k}\) b. \(\sum_{k=0}^{\infty} 0.15^{k}\)

Consider the following infinite series. a. Write out the first four terms of the sequence of partial sums. b. Estimate the limit of \(\left\\{S_{n}\right\\}\) or state that it does not exist. $$\sum_{k=1}^{\infty} \cos (\pi k)$$

Consider the following infinite series. a. Write out the first four terms of the sequence of partial sums. b. Estimate the limit of \(\left\\{S_{n}\right\\}\) or state that it does not exist. $$\sum_{k=1}^{\infty} 9(0.1)^{k}$$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.