Download key - BetsyMcCall.net

Math 266, Quiz #4, Summer 2010
Name _______________________________________________
Instructions: Show all work. On proofs, clearly explain your reasoning. Unexplained leaps of logic, even
if correct, will be treated as if it is false. On take home quizzes, all work must be your own; you may not
work together.
1. List the first four terms of each sequence. Write a rule for the general term.
a. The sequence obtained by starting with 10 and each successive term by subtracting 3 from
the previous term.
10,7,4,1,…
an = 10-3n for n=0,1,2,3… or an=13-3n for n=1,2,3… or an=an-1 -3
b. The sequence whose nth term is the sum of the first n positive integers.
n
1,3,6,10,…
an=
i
i 1
or an=an-1 + n
c. The sequence whose nth term is 3n-2n
(0), 1, 5, 19, 65,…
(an given)
2. Find the domain and range of these functions.
a. The function that assigns to each pair of positive integers the first integer of the pair
f : Z Z  Z
b. The function that assigns to each positive integer its largest decimal digit.
f : Z   {0,1,2,3,4,5,6,7,8,9}
c. The function that assigns each positive integer the largest integer not exceeding the square
root of the integer.
f : Z  Z
3. What is the value of each of these sums of terms of a geometric progression?
8
a.
3 2
j
1533
j0
8
b.
 ( 3)
i 2
i
4923
4. Determine whether each of these sets is countable or uncountable. For those that are
countable, exhibit a one-to-one correspondence between the set of natural number and that
set.
a. Integers not divisible by 3
countable
x
f(x)
1
1
2
2
4
3
5
4
7
5
8
6
10
7
11
8
13
9
25
5
30
6
40
7
45
8
50
9
1.11
8
.111
9
b. Integers divisible by 5 but not by 7
countable
x
f(x)
5
1
10
2
15
3
20
4
c. The real numbers with decimal representations consisting of all 1s
countable
x
f(x)
1
1
.1
2
11
3
1.1
4
.11
5
111
6
11.1
7
d. The real numbers with decimal representations of all 1s or 9s
Countable (I interpret this to be all (1s or 9s) but if you interpret this to be all 1s or all 9s then the set is
just like the one for c, but let the odds be the 1’s and the evens the 9s)
x
f(x)
1
1
9
2
.1
3
.9
4
11
5
19
6
91
7
99
8
1.1
9