   Chapter 11.R, Problem 9E

Chapter
Section
Textbook Problem

A sequence is defined recursively by the equations a 1 = 1 , a n + 1 = 1 3 ( a n + 4 ) . Show that { a n } is increasing and a n < 2 for all n. Deduce that { a n } is convergent and find its limit.

To determine

To show:

i) The sequence  an is increasing, and an<2, n.

ii) Deduce that  an is convergent and find its limit.

Explanation

1) Concept:

i) If limnan exists, then the sequence  an converges; otherwise, the sequence diverges.

ii) A sequence an is called increasing if  an<an+1 for all n1.

iii) If limnan+1=L,  then limnan=L.

2) Given:

a1=1, an+1=13an+4, for n1

3) Calculation:

Consider the given sequence

an+1=13an+4

To prove  an  is increasing:

That is, to prove  an-1<an<2 by using Mathematical induction method.

It is given that a1=1

a2=13a1+4

=131+4

a2=53<2

This is true for n=2.

Assume that it is true for n=k.

ak-1<ak<2

Now to prove that it is true for n=k+1, add 4 to each term in the above inequality.

ak-1+4<ak+4<2+4

Now multiply all these terms by 13.

13ak-1+4<13ak+4<13(6)

That is,

ak<ak+1<2

This is true for n=k+1

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Find more solutions based on key concepts 