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The Nuts and Bolts of Proofs, 4th Edition
book

The Nuts and Bolts of Proofs, 4th Edition

by Antonella Cupillari
January 2011
Beginner content levelBeginner
296 pages
11h 43m
English
Academic Press
Content preview from The Nuts and Bolts of Proofs, 4th Edition
This statement is false since it implies that 4 < 3. Therefore the statement that t here exists a number
k 1forwhicha
k
< a
k+1
is false. So we can concl ude that a
n
> a
n+1
for all n and the sequenc e is
decreasing.
Since the sequence is decreasing, by the result in Exercise 1, a
n
< a
1
for all n 2. The sequence is
also positive. Thus 0 a
n
< a
1
= 1:5foralln 2 . So the se quence is bounded by M = 1.5(andany
number larger than or equal to 1.5).
A possible recursive formula for the sequence is a
n+1
= a
n
1
ðn + 1Þðn + 2Þ
with a
1
= 1:5
no
n=1
:
5. We can start by finding the values of a few terms of the sequence:
a
1
= 0 a
2
= 1 a
3
= 0
a
4
= 1 a
5
= 0 a
6
= 1
The sequence
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Publisher Resources

ISBN: 9780123822178