Math 216 Exam 3 (150 points) Name _____________________
Homework points (max of 50)
________
REMOVE THE
LAST PAGE – FOR USE AS A REFERENCE SHEET
1. Questions on recursively
defined sequences.
a. Write the first four terms of the following
sequence: (4 points)
b. Write the first five terms of the
following sequence: (5 points)
2. Recall the definition of the
Fibonacci numbers. (4 points)
Using the definition, show that
3. Suppose a certain amount of
money is deposited into an account paying 6% annual interest compounded quarterly. For each positive integer n, let Rn = the amount on deposit at the end of the nth quarter, assuming no
additional deposits or withdrawals, and let R0
be the initial amount deposited.
a. Write the recurrence
relation for Rn in terms
of Rn-1. (3 points)
Rn =
b. If R0 = $10,000, find the amount of money on deposit at the
end of a year, that is R4.
(4 points)
(may be done with or without reference sheet)
c. What is the APR (annual
percentage rate) for the account? (2
points)
4. Santa is in a bind. Because of the expanding population, his
elves will need to produce more dolls this year than ever before. He has promised extra cookies to any elf that
can increase his daily production by 3 dolls per day for each of the next 25
days. If an elf has been producing 200
dolls per day, how many dolls must he be producing on the 25th day
to get the extra cookies? (4 points)
(reference sheet should be helpful)
5. This puts Mrs. Claus in a
bind as well. She has to make more
cookies this year than ever before. Her
plan is to increase her daily cookie production by 2% each day for the next 25
days. If she has been baking 1,000
cookies per day, how many cookies will she be making on the 25th day
if she increases at 2% each day? (4
points)
(reference sheet should be helpful)
6. Let A = {2, 3, 4} and B = {8,
10, 12} and define the divides binary
relation R on A to B as follows:
a. Is 2 R 8? (1 point)
b. Is 4 R 10? (1 point)
c. Is 2 R 4? (1 point)
d. Is (3,12) Î R? (1 point)
e. Write R as a set of ordered pairs.
(3 points)
f.
Draw the arrow diagram for R. (3 points)
A B
7. Circle Y if a relation must have that property to be classified as the
given type of relation, N
otherwise. (8 points)
|
type |
reflexive |
symmetric |
antisymmetric |
transitive |
|
equivalence
relation |
Y or N |
Y or N |
Y or N |
Y or N |
|
partial
order |
Y or N |
Y or N |
Y or N |
Y or N |
8. Consider the following
relations on the set {1, 2, 3, 4}. For
each property, circle Y if that
relation has that property, N
otherwise. (12 points)
R1 = {(1,2), (2,3), (3,4),
(4,1), (2,1), (3,2), (4,3), (1,4)}
R2 = {(1,1), (1,2), (2,2),
(3,3), (3,2), (2,3), (4,4)}
R3 = {(1,1), (2,2), (3,3),
(4,4)}
|
relation |
reflexive |
symmetric |
antisymmetric |
transitive |
|
R1 |
Y or N |
Y or N |
Y or N |
Y or N |
|
R2 |
Y or N |
Y or N |
Y or N |
Y or N |
|
R3 |
Y or N |
Y or N |
Y or N |
Y or N |
9. Consider the following
infinite relations. For each property,
circle Y if that relation has that
property, N otherwise. (12 points)
G is
the “greater than or equal to” relation on the set of real numbers R
as follows:
For
all x,yÎ R, x G
y Û x ≥ y.
D is
the binary relation defined on the set of real numbers R as follows:
For
all x,yÎ R, x D
y Û xy ≥ 0.
F is
the congruence modulo 5 relation defined on the set of integers Z
as follows:
For
all m,nÎ Z, m F
n Û 5 | m - n.
|
relation |
reflexive |
symmetric |
antisymmetric |
transitive |
|
G |
Y or N |
Y or N |
Y or N |
Y or N |
|
D |
Y or N |
Y or N |
Y or N |
Y or N |
|
F |
Y or N |
Y or N |
Y or N |
Y or N |
10. The relation F from the previous question is an equivalence relation with 5
equivalence classes. They are listed below. For each, give one integer other than 1, 2,
3, 4, or 5 that belongs to the equivalence class. (5 points)
|
equivalence class |
one integer other than 1, 2, 3, 4, or 5 in this class |
|
[1] |
|
|
[2] |
|
|
[3] |
|
|
[4] |
|
|
[5] |
|
11. The divides relation on the set A = {1, 2, 4, 5, 10, 15, 20} is a partial order:
a. Draw the Hasse diagram. (4 points)
b. Find all greatest, least, maximal, and minimal elements. If there is none, write “none.” (4 points)
greatest:
least:
maximal:
minimal:
12. Solve the following
recurrence relation using the 3-step
method of iteration, guessing and proof using mathematical induction. (15 points)
Reference Sheet
