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158
Solutions to Review Exercises
which is indeed true. The induction step is complete; we conclude that (1 
~)
n
;:::
1 
~
for all
n;:::1.
4. When
n
= 1, the sum is 1 > 2(v'2 
1) because v'2
~
1.4. Now assume that
k
> 1 and that
L~=1 ~
>
2(
y'kTI
1). We have
k+l
1
k
1
1
2:=2:+=
i=1
Vi
i=1
Vi
y'kTI
>2(Vk+l1)+
~
yk+1
2(k
+
1) 
2y'kTI
+
1
y'kTI
by the induction hypothesis
We want to show that this quantity is at least as large as
2Jk
+
2 
2, in order to get the desired
inequality with
n
=
k
+
1. Now
2(k
+
1) 
2y'kTI
+
1
>
2Vk+2 _ 2
Vk+1 
+
if and only if
2(k
+
1) 
2Vk+1
+
1;::: 2v'k"TIVk"+2 
2y'kTI
if and only if
2(k
+ 1) + 1 ;:::
2Jk2
+
3k
+ 2
if and only if
2k
+ 3;:::
2Jk2
+
3k
+ 2
and this is true because the numbers on each side of this inequality are positive and
(2k
+
3)2
4k2
+
12k
+ 9 >
4(k
2
+
3k
+ 2). We have established the inequality for
n
=
k
+ 1 and conclude, by
the Principle of Mathematical Induction, that it holds for all
n
;::: 1.
5. When
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This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.
 Summer '10
 any
 Graph Theory

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