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SIMON FRASER UNIVERSITY
DEPARTMENT OF MATHEMATICS
Midterm 2
MACM 201 Spring 2008 Instructor: Robert ˇS´amal March 5, 2008, 12:30 – 13:
Name: (please print) family name given name
SFU ID: student number SFU-email
Signature:
Instructions:
- Do not open this booklet until told to do so.
- Write your name above in block letters. Write your SFU student number and email ID on the line pro- vided for it.
- Write your answer in the space provided below the question. If additional space is needed then use the back of the previous page. Your final answer should be simplified as far as is reasonable.
- Make the method you are using clear in every case unless it is explicitly stated that no explanation is needed.
- This exam has 5 questions on 5 pages (not includ- ing this cover page). Once the exam begins please check to make sure your exam is complete.
- No calculators, books, papers, or electronic devices shall be within the reach of a student during the examination. The only exception is your formula sheet—a one-sided sheet of paper.
- During the examination, communicating with, or deliberately exposing written papers to the view of, other examinees is forbidden.
Question Maximum Score
Total 50
[10] 1. Solve the following recurrence relation
an+1 + 2an = 2n^ (n ≥ 0), a 0 = 1.
You may use any of the methods we learned.
3. Find generating functions for the following sequences (include all the necessary compu-
tation).
[1] (a) 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 ,...
[2] (b) 1 , − 1 , 1 , − 1 , 1 , − 1 ,...
[2] (c) 2 , 0 , 4 , 0 , 8 , 0 , 16 ,...
[3] (d) The generating function for the sequence of third powers, 03 , 13 , 23 , 33 , 43 , 53 ,...
is f (x) =
x^3 + 4x^2 + x (1 − x)^4
. (You don’t need to verify this.) Use this fact to find the generating function for 0 , 0 , − 13 , 23 , − 43 , 53 ,...
[2] (e) Find [x^2 ]( (^2) x + 3x)^10.
[4] (f) Find [x^60 ]1+3 (1−xx+ (^3) )x 32.
4. We want to distribute 101 cookies to three children: Adam, Beata, and Colin.
- Adam always eats two cookies at the same time, so we want to give him an even number of cookies, but not more than 20.
- Beata has a birthday today, so we want to give her at least 20 cookies.
- Colin can get any number of cookies.
Find out how many ways we have to distribute the cookies.
[3] (a) Express the answer as a coefficient of some power of x in an appropriate generating function f (x). (Don’t forget to explain why your formula is correct.)
[4] (b) Express f (x) as a polynomial (1−x (^2) ) 3.
[3] (c) Use the binomial theorem to get the final answer.
