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) 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,y R, x G y xy.

 

D is the binary relation defined on the set of real numbers R as follows:

For all x,y R, x D y xy ≥ 0.

 

F is the congruence modulo 5 relation defined on the set of integers Z as follows:

For all m,n Z, m F n 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