   Chapter 11.1, Problem 82E

Chapter
Section
Textbook Problem

Show that the sequence defined by a 1 = 2 → a n + 1 = 1 3 − a n satisfies 0 < a n ≤ 2 and is decreasing. Deduce that the sequence is convergent and find its limit.

To determine

To show:

(i) The sequence satisfies 0<an2 and is decreasing. Deduce that an is convergent.

(ii) Find its limit.

Explanation

1) Concept:

Use induction to show an is decreasing and satisfies 0<an2. Use monotonic convergence theorem to show limnan exists. Using the limit laws, find the limit.

2) Definition:

Monotonic Sequence Theorem:

Every bounded, monotonic sequence is convergent.

3) Given:

a1=2    an+1=13-an

4) Calculation:

(i)

Begin by computing the first several terms.

a1=2

an+1=13-an

a2=a1+1=13-a1=13-2=11=1

a3=a2+1=13-a2=13-1=12=0.5

a4=a3+1=13-a3=13-1/2=26-1=25=0.4

a5=a4+1=13-a4=13-2/5=513=0.384

a6=a5+1=13-a5=13-5/13=1334=0.3823

a7=a6+1=13-a6=13-13/34=3489=0.38202

a8=a7+1=13-a7=13-34/89=89267=0.3333

a9=a8+1=13-a8=13-89/267=267801=0.3333

These initial terms suggest that the sequence is decreasing and the terms are approaching 0.3333,

and it satisfies 0<an2

To confirm that the sequence is decreasing, use mathematical induction to show that

an+1<an for all n1

This is true for n=1 because a2=1<a1=2

Assume that it is true for n=k, then

ak+1<ak

Multiply by (-) sign.

-ak+1>-ak

Add 3 on both sides,

3-ak+1>3-ak

Taking reciprocal,

13-ak+1<13-ak

ak+2<ak+1

Therefore,

ak+1<ak is true for n=k+1

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