Chapter 9, Problem 10PS

### Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

Chapter
Section

### Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem

# Proof(a) Consider the following sequence of numbers defined recursively. a 1 = 3 a 2 = 3 a 3 = 3 + 3               ⋮ a n + 1 = 3 + a n Write the decimal approximations for the first six terms of this sequence. Prove that the sequence converges and find its limit.(b) Consider the following sequence defined recursively by a 1 = a   and   a n + 1 = a + a n ,where a > 2 . a , a + a , a + a + a , ⋯ Prove that this sequence converges and find its limit.

(a)

To determine

To prove: The given sequence a1=3a2=3a3=3+3an+1=3+an converges.

Explanation

Given:

The given sequence is:

a1=3a2=3a3=3+3an+1=3+an

Proof:

Consider the provided sequence:

a1=3a2=3a3=3+3an+1=3+an

Now, the nth term of the provided sequence is:

an+1=3+an

Therefore, the first six terms of this sequence are,

a1=3.0a2=31.73205a3=3+32.17533a4=3+a3=3+2.175332.27493

And,

a5=3+a4=3+2.274932.29672a6=3+a5=3+2.296722.30146

Suppose the sequence increases that is,

a2=a+a1=aa>a=a1

So, assume an>an1

(b)

To determine

To prove: The given sequence a,a+a,a+a+a, converges.

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