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Exercise
Set
5.2
1.
Use mathematical induction (and the proof of Proposi
tion 5.2.1 as a model)
to
show that any amount of money
of at least 14¢ can be made up using 3¢ and 8¢ coins.
2.
Use mathematical induction to show that any postage of at
least 12¢ can be obtained using 3¢ and 7¢ stamps.
3. For each positive integer
n,
let
pen)
be the formula
2
2
2
n(n
+
1)(2n
+
I)
1+2+···+n= 6
.
a.
Write
P(l).
Is
P(1)
true?
b.
Write
P(k).
c.
Write
P(k
+
I).
d.
In a proof by mathematical induction that the formula
holds for all integers
n
2: I, what must be shown in the
inductive step?
4.
For each integer
n
with
n
2: 2, let
pen)
be the formula
1/1
"
..
n(nI)(n+l)
8
1
(1+1)= 3
.
a.
Write
P(2).
Is
P(2)
true?
b.
Write
P(k).
c.
Write
P(k
+
I).
d.
In a proof by mathematical induction that the formula
holds for all integers
n
2: 2, what must be shown in the
inductive step?
5. Fill in Ole missing pieces in the following proof that
I + 3 + 5 + .
.. +
(2n

I) =
n
2
for all integers
n
2: I.
Proof: Let the property
pen)
be the equation
2
I + 3 + 5 + .
.. +
(2n

I) =
n
.
~
P(n)
Show that P(l)
is true: To establish
P(l),
we must show
that when I is substituted in place of
n,
the lefthand side
equals the righthand side. But when
n
=
I, the lefthand
side is the sum of all the odd integers from I to 2· I 
I,
which is the sum of the odd integers from I to I, which is
just I. The righthand side is
~,
which also equals I. So
PO)
is true.
Show that for all integers k
~
1,
if
P(k)
is
true then
P(k
+
1)
is true:
Let
k
be any integer with
k
2: 1.
[Suppose P(k) is true. That is:]
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This note was uploaded on 06/12/2011 for the course MATH 103 taught by Professor Wouters during the Spring '08 term at Wisc Oshkosh.
 Spring '08
 WOUTERS
 Algebra, Mathematical Induction

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