Here is a list of questions I have asked on previous exams pertaining to topics
covered so far in this course as well as some new questions I have created. Note
that just because a topic from those sections is not represented here, does
not
mean it will not be tested on. This list is simply meant to be representative.
•
Solve the following recurrence relation
a
n
= 5
a
n

1

6
a
n

2
+ (7

2
n
)2
n

1
a
0
= 2
a
1
= 1
.
•
Let
a
n
= 4
a
n

1
+ 8
a
n

2
+ 7
a
n

3
, with
a
0
= 0,
a
1
= 1, and
a
2
=

1 Find
the generating function for
a
n
.
•
Solve
a
n
= 3
a
n

1

2
a
n

2

n
2
n
when
a
0
=

2 and
a
1
=

1.
•
Find generating functions for
a
n
and
b
n
, where
a
0
= 1,
b
0
= 3, and
a
n
=
a
n

1

3
b
n

1
b
n
=

a
n

1
+ 2
b
n

1
.
•
Find
a
n
for
n
≥
2, when
a
2
= 4
, a
3
= 5 and
a
n
=
a
n

1
+ 6
a
n

2
,
for
n
≥
4.
•
Evaluate the sum
∑
54
i
=0
∑
54

i
j
=0
∑
54

j

k
k
=0
54!
i
!
j
!
k
!(54

i

j

k
)!
2
i
√
3
j
(

3)
54

i

j

k
•
Prove that for all
n
≥
0,
∑
(
n
+1)
2
k
=
n
2
+1
k
=
n
3
+ (
n
+ 1)
3
.
•
Determine the number of positive integers solutions to
x
1
+
x
2
+
x
3
+
x
4
=
17, where
x
1
∈ {
1
,
2
,
3
}
, 4
≤
x
2
, 0
≤
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 Spring '09
 COSTELLO
 Math, Natural number, Bachelor's degree, Recurrence relation, Generating function

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