MACM 201 Midterm 3 Practice Problems
Mahsa Faizrahnemoon, Michael Monagan and Amarpreet Rattan
1. For each of the following equations determine a generating function A(x) for the number of
integer solutions. Indicate the coefficient of A(x) needed to solve the problem.
(a) c1 + c2 + c3 + c4 = 10 for 1 ≤ ci ≤ 4.
(b) 2c1 + c2 + c3 + c4 = n for 1 ≤ ci .
(c) c1 + c2 + c3 = 20 for 0 ≤ ci and c1 is odd.
2. Find the coefficient of x50 in (x6 + x7 + x8 + x9 + . . . )6 .
Express your answer in calculator ready form.
3. In how many ways can Mary divide 12 hamburgers and 16 hotdogs among her sons Alex, Bill
and Chris such that Alex and Bill get at least one hamburger and 1 hotdog each and Chris
gets at least 2 hamburgers. Use generating functions.
4. Give a rational generating function for the sequences
(a) 1, 3, 9, 27, 81, 243, . . .
(b) 0, 1, 2, 3, 4, 5, . . .
(c) 2, −2, 2, −2, 2, −2, . . .
5. Use partial fractions to find a formula for [xn ]A(x) where A(x) =
x
.
(1 − x)(1 − 2x)2
You should get (n − 1)2n + 1.
6. Consider the recurrences
(a) an = 2an−1 + n for n ≥ 1 with a0 = 1.
(b) an = 2an−1 − an−2 + 1 for n ≥ 2 with a0 = 1 and a1 = 1.
Find a rational generating function that generates the sequence a0 , a1 , a2 , . . . .
You don’t need to determine a formula for the coefficient of xn .
7. Show that
n
X
k=0
k=
n(n + 1)
for n ≥ 0 by induction on n.
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8. The Fibonacci generating function F (x) is x/(1−x−x2 ). List the first 6 terms of F (x)/(1−x).
9. (a) What is the difference between a path and a trail?
(b) What is an Euler circuit? Draw the smallest multi-graph which has an Euler circuit.
(c) What is an Euler trail? Draw the smallest multi-graph which has an Euler trail.
10. An exam question I found on the internet reads: “If G is a graph with one vertex of odd
degree can it have an Euler trail?” What is wrong with this question?
11. Use series division to determine the coefficients of the following series up to x4 . Furthermore,
find a general recurrence for the nth coefficient of the series. State enough initial conditions
for the recurrence so that the recurrence determines a unique sequence.
1
1+x
.
1 − x − x3
1−x
(b)
.
1 − x2 − x3
P
12. Prove that nk=1 k(k + 1) = 2
(a)
n+2
3
for all n ≥ 1 by induction.
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