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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
(Continued )
If we are considering an element with an even index, it will be an element of the kind a
k
= ð1Þ
k
k + 1
k + 2
where k =2t, with a natural number. Thus,
a
2t
= ð1Þ
2t
2t + 1
2t + 2
=
2t + 1
2t + 2
:
The next term in the sequence with an even index will be a
2t +2
, and
a
2t + 2
= ð1Þ
2t + 2
ð2t + 2Þ+ 1
ð2t + 2Þ+ 2
=
2t + 3
2t + 4
:
Lets prove the claim that a
2t
is smaller than a
2t +2
by checking the difference a
2t +2
a
2t
:
a
2t + 2
a
2t
=
2t + 3
2t + 4
2t + 1
2t + 2
=
2
ð2t + 4Þð2t + 2Þ
:
This fraction is always positive. Thus, a
2t
is smaller than a
2t +2
, but a
2t
is larger than a
2t+1
. So, the
sequence is not monotone. On the other hand, the sequence is bounded because
j
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Publisher Resources

ISBN: 9780123822178