Study Guide
Math 366
Test II
You should know the material we have covered since the last test, starting with gener-
ating functions and recurrence relations (chapter 6 and 7). You should know the standard
generating functions and be able to apply them in solving problems. You should be able
to find the generating function for counting problems. Moreover, you should be able to
use a generating function to actually find the numbers for such problems. In addition,
you should be able to use recurrence relations to solve counting problems. Moreover,
you should be able to solve linear recurrence relations using the characteristic polynomial
and finding roots. You should also be able to solve recurrence relations with generating
functions. In general, you should be able to solve any problem that has been assigned on
the homework as well similar types of problems.
The test will come in two portions, a calculator/study sheet free portion and a portion
where you can use a calculator and study sheet. The study sheet should be one standard
size piece of paper. On the no study sheet portion, you will be expected to state definitions
and some generating functions from memory. In addition, you will need to solve some
problems by hand. Specific items that you will be expected to state from memory include
(but are not limited to)
•The generating function expansions for
1. (1 + ax)n= 1 + n
1ax +n
2a2x2+· · · +n
nanxn=
n
X
k=0 n
kakxk.
2. 1
1−ax = 1 + ax +a2x2+a3x3+. . . .
3. 1
(1−x)2= 1 + 2x+ 3x2+. . . .
•Generating functions for standard counting problems.
•The definition of a linear recurrence.
•The characteristic polynomial for a linear recurrence.
•How to apply recurrences to standard counting problems.