MATH 135
Fall 2007
Assignment #2
Due: Wednesday 26 September 2007, 8:20 a.m.
N.B. Assignments 3 to 9 will not be distributed in class. You must download them from the course
Web site.
HandIn Problems
1. A sequence of integers is defined by
x
1
= 2,
x
2
= 86,
x
m
+2
=

2
x
m
+1
+ 15
x
m
for
m
≥
1. Prove
that
x
n
= 4
·
3
n
+ 2(

5)
n
for all
n
∈
P
.
2. A sequence of integers is defined by
y
1
= 2,
y
2
= 11,
y
3
= 2,
y
m
+3
= 3
y
m
+1

2
y
m
for
m
≥
1.
Prove that
y
n
= (

2)
n
+ 3
n
+ 1 for all
n
∈
P
.
3. Suppose that
x, y
∈
R
. Prove that
x
2

4
xy
+ 5
y
2

4
y
+ 4 = 0 if and only if (
x, y
) = (4
,
2).
4. Prove by contradiction that log
10
3 cannot be written in the form
m
n
where
m
and
n
are positive
integers.
5. Prove that if
x
=
y
and
a
= 0, then
x
y
=
x
+
a
y
+
a
.
6. To prove a statement of the form “If
H
then
C
1
or
C
2
” (where
H, C
1
, C
2
are mathematical
statements), we can assume that
H
is TRUE and that
C
1
is FALSE, and prove that
C
2
is
TRUE. (If
C
1
happened to be TRUE, we would be done, so we assume that
C
1
is FALSE and
prove that
C
2
must be TRUE.)
Suppose that
m
and
n
are integers.
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 Winter '08
 ANDREWCHILDS
 Integers, Degree of a polynomial, Mathematical terminology, mathematical statements

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