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
6
=
y
and
a
6
= 0, then
x
y
6
=
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.
(a) Prove that if 3
m
2
+ 6
m
+ 5
n
2
+ 3 is odd, then
m
is even or
n
is odd.
(b) Is the converse TRUE or FALSE? Either prove or give a counterexample.
7. An expression of the form
a
n
x
n
+
a
n

1
x
n

1
+
···
+
a
1
x
+
a
0
with
n
≥
0 and
a
n
, a
n

1
, . . . , a
1
, a
0
∈
Q
(or
R
) is called a polynomial in
x
with coeﬃcients from
Q
(or
R
).
If
a
n
6
= 0, the polynomial is said to have degree
n
. (The degree of a polynomial is the highest
power of
x
that has a nonzero coeﬃcient.) If all of the coeﬃcients are 0, the polynomial is
called the zero polynomial and its degree is not deﬁned.
(a) In each part, state whether the expression is a polynomial. If it is a polynomial, state its
degree.
i.
√
2
x
3

πx
+ 3
ii. 2(
√
x
)
3

πx
+ 3
iii. 4
(b) In each part, perform the calculation, simplifying as much as possible.
i. (
x
2

√
2
x
+ 1)(
x
2
+
√
2
x
+ 1)
ii. (
x
2
+ 3
x
+ 1)(
x
3
+
x
2

2) + 3
x
4
+ 4
x
2
+ 5
...continued