Due Friday, January 22, 2010
Give rigorous solutions of all problems.
Problem 1.
Describe all integer solutions of the equation
n
2
+ 2
m
2
=
l
2
Hint: see Handout 1.
Answer:
n
= 2
s
2

t
2
,m
= 2
st,l
= 2
s
2

t
2
where
s,t
∈
Z
.
Problem 2.
Prove, using mathematical induction that
1 + 3 + 5 + 7 +
···
+ 2
n

1 =
n
2
Proof. We will prove this formula by induction. First, we check the base of
induction:
Base of induction:
1 = 1
2
Step of induction: Assume for some number
n
the formula is correct:
1 + 3 + 5 + 7 +
···
+ 2
n

1 =
n
2
We need to prove that
1 + 3 + 5 + 7 +
···
+ 2
n

1 + (2
n
+ 1) = (
n
+ 1)
2
In order to do it we compare how the right and left sides of the formula
change when we go from
n
to
n
+ 1
.
Left side changes by
2
n
+ 1
.
Right side changes by
(
n
+ 1)
2

n
2
=
n
2
+ 2
n
+ 1

n
2
= 2
n
+ 1
.
Both sides of the formula change in the same way. Therefore, if the formula
is true for
n
then it is true for
n
+ 1
.
1
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 Spring '08
 MULLEN
 Math, Mathematical Induction, Natural number, Leftwing politics, Mathematical logic, Mathematical proof

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