   Chapter 11.1, Problem 80E

Chapter
Section
Textbook Problem

# A sequence {an} is given by a 1 = 2 , a n + 1 = 2 + a n . (a) By induction or otherwise, show that {an} is increasing and bounded above by 3. Apply the Monotonic Sequence Theorem to show that limn→∞ an exists. (b) Find limn→∞ an.

(a)

To determine

To show: The sequence {an} is increasing and bounded above by 3 and limnan exists.

Explanation

Given:

The sequence {an} is a1=2, an+1=2+an (1)

Theorem used: Monotonic Sequence Theorem

“If the sequence is bounded and monotonic, then the sequence is convergent.”

Proof: Induction method

Claim: Let Pn be the statement that an+1an and an3 . (2)

Base case: n=1

Substitute 1 for n in equation (2),

a1+1a1 and a13a2a1 and a13

Obtain the value of a2 .

Substitute 1 for n in equation (1),

a1+1=2+a1a2=2+2 [a1=2]=1.848

Thus, the value of a2=1.848 .

Since a2=1.848 and a1=1.414 , a2a1  and  a13 .

Thus, the claim is true for n=1 .

Induction hypothesis: n=k

Assume that the claim is true when n=k .

Let Pk be the statement that ak+1ak and ak3 .

Inductive step: n=k+1

To prove that the claim is true when n=k+1

(b)

To determine

To find: The limit of the sequence.

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