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FUNDAMENTALS OF ALGEBRA
VAN WYK’S MATH 304
1. Number Patterns (Last updated 5/3/11)
1.1. Number Systems. You have probably seen the following sets of numbers:
(1) The integers, Z = {. . . , −3, −2, −1, 0, 1, 2, 3, . . . }.
(2) The rationals, Q, each of which can be expressed as a quotient of two integers.
(3) The reals, R, which lie in one-to-one correspondence with the points on the number
line.
Each set above is a subset of the next. In symbols, Z ⊆ Q ⊆ R. The numbers that are
elements of R but not of Q are known as irrationals.
You have probably also heard each of the following terms as well: counting numbers,
natural numbers, and whole numbers. Unfortunately, each of these can mean either
{0, 1, 2, 3, . . . } or {1, 2, 3, 4, . . . }, so you have to be careful when reading or using these
terms. You can avoid confusion by using the terms positive integers, often denoted Z+ ,
and the non-negative integers, which consist of the positive integers and zero.
Example 1.1. The number −7 is an integer, so it is also a rational and a real. The number
2
7
−4
is a rational (and a real), but not an integer; it can also be expressed as −14
, or in infinitely
many other ways as a quotient of two integers. The positive number whose square is two,
√
2, is a real number that is not rational (thus is irrational), as it cannot be expressed as a
quotient of two integers, although this is not obvious.
You may have also heard of the complex numbers, C, each of which can be expressed
√
uniquely in the form a + bi, where a and b are real numbers and i represents −1. We will
be focusing on the reals and subsets of the reals in this course, so when we use the word
number, you can assume we are referring to a real number, not a complex number.
1.2. Sequences.
Definition 1.1. A sequence of numbers is an ordered list of numbers. The numbers
themselves are called terms of the sequence.
The numbers are usually separated by commas. The list might be finite or infinite. The
numbers might be all different, or all the same, or neither. The standard notation for the
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VAN WYK’S MATH 304
general form of a sequence is
a1 , a2 , a3 , a4 , . . .
In this case, the full sequence itself is often denoted (an ).
Example 1.2. The following are sequences:
(1) 2, 4, 8, 14, 22
(2) 2, 4, 8, 14, 22, 32, 44, 58, 74, 92, 112
(3) 2, 4, 8, 14, 22, 32, . . .
(4) 21 , 32 , 34 , 54 , . . .
Notice the first two sequences are finite; the first has length 5, while the second has length
11. In order to use elipses (. . . ) in the definition of a sequence, as in the last three examples,
there must be a discernible pattern to the sequence1. Notice the terms of the first three
sequences are given by adding increasing multiples of 2 to the previous term, starting with
the number 2. (Thus the second term of the sequence is 2+2 = 4, the third term is 4+4 = 8,
the fourth is 8 + 6 = 14, etc.) You should be able to identify the pattern in the last sequence.
Being able to define (possibly infinite) sequences of numbers efficiently is a handy skill.
Not all sequences will submit themselves to such definitions – for example, a sequence of
randomly-chosen real numbers – but many sequences of interest do. For those sequences for
which each term depends on previous terms, we can define them in the following way.
Definition 1.2. A recursive definition of a sequence of numbers consists of two steps:
Base Step: Give the first term (or possibly the first few terms) of the sequence.
Recursive Step: Give a rule for finding each term of the sequence from previous terms.
The basic idea is that once you know the first term, you can use the recursive step to
compute the second term. Then you can use the second term and the recursive step to
compute the third term, etc.
Example 1.3. Let’s try a recursive definition for the sequence in Example 1.2 (3). Call the
full sequence (an ). We need to find a pattern. Notice
a1 = 2
a2 = 4 = 2 + 2 · 1
a3 = 8 = 4 + 2 · 2
a4 = 14 = 8 + 2 · 3
1A
a5 = 22 = 14 + 2 · 4
word of caution is in order here. Any finite sequence can be extended to an infinite sequence in
infinitely many ways that preserve the initial pattern. So when we say discernible pattern, we loosely mean
the most obvious pattern.
FUNDAMENTALS OF ALGEBRA
3
Thus each term is the sum of the previous term and two times the previous subscript. So
a recursive definition of (an ) is
Base Step: a1 = 2.
Recursive Step: an = an−1 + 2 · (n − 1) (for n ≥ 2).
Starting from this recursive definition, you should recreate the first six terms of the sequence.
Example 1.4. The famous Fibonacci numbers are the terms of the sequence (fn ), given by
the following recursive definition:
Base Step: f0 = 0 and f1 = 1.
Recursive Step: fn = fn−1 + fn−2 (for n ≥ 2).
The recursive step indicates each term is the sum of the previous two terms. Thus the terms
of the Fibonacci sequence are:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, . . .
Some sources begin the Fibonacci sequence at f1 ; in this case, the base step consists of
f1 = f2 = 1, while the recursive step remains the same but only applies for n ≥ 3.
The Fibonacci sequence is rich with properties, as you will see with an internet search.
1.3. Arithmetic and Geometric Sequences. Two types of common sequences that can
be defined recursively are arithmetic2 sequences and geometric sequences. The terms of the
former are given by adding a fixed number to the previous term, while the terms of the
latter are given by multiplying the previous term by a fixed number . In other words, two
successive terms in an arithmetic sequence have a common difference, while two successive
terms in a geometric sequence have a common ratio.
Definition 1.3. An arithmetic sequence (an ) with common difference d and initial term
a is defined recursively by
Base Step: a1 = a.
Recursive Step: an = an−1 + d (for n ≥ 2).
A geometric sequence (an ) with common ratio r and initial term a is defined recursively
by
Base Step: a1 = a.
Recursive Step: an = an−1 · r (for n ≥ 2).
Example 1.5. 7, 4, 1, −2, −5, . . . is an arithmetic sequence with common difference d = −3,
, 20 , − 40
, . . . is a geometric sequence with common ratio r = − 23 .
while 5, − 10
3 9
27
2Here
“arithmetic” is pronounced ar-ith-met-ik, not uh-rith-muh-tik.
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VAN WYK’S MATH 304
There are two properties that both arithmetic sequences and geometric sequences possess:
(1) It is easy to determine the nth term of the sequence without computing all the
previous terms.
(2) It is easy to compute the sum of the first n terms.
If you try to compute either of these two things for the sequences in Example 1.2 (3) or
Example 1.4, you might appreciate these two properties.
Starting with an arithmetic sequence (an ) with initial term a and common difference d,
we have the following:
a1 = a
a2 = a1 + d = a + d
a3 = a2 + d = (a + d) + d = a + 2d
a4 = a3 + d = (a + 2d) + d = a + 3d
a5 = a4 + d = (a + 3d) + d = a + 4d
Thus it appears that the nth term of an arithmetic sequence is an = a + (n − 1)d. In fact,
this can be rigorously proven using the Principle of Mathematical Induction3.
Now, starting with a geometric sequence (an ) with initial term a and common ratio r, we
have the following:
a1 = a
a2 = a1 · r = a · r
a3 = a2 · r = (a · r) · r = a · r 2
a4 = a3 · r = (a · r 2 ) · r = a · r 3
a5 = a4 · r = (a · r 3 ) · r = a · r 4
Thus it appears that the nth term of a geometric sequence is an = a · r n−1 . This can also be
proven using the Principle of Mathematical Induction.
1.4. Sums of Terms of Arithmetic and Geometric Sequences. Turning to sums of
the terms in these sequences, lets first look at an example.
Example 1.6. The first seven terms from the arithmetic sequence with initial term 7 and
common difference −3 from Example 1.5 are 7, 4, 1, −2, −5, −8, −11. Let’s add the terms in
3This
is stated in Section 2.
FUNDAMENTALS OF ALGEBRA
5
pairs, working from the outside in:
a1 + a7 = 7 + (−11) = −4
a2 + a6 = 4 + (−8) = −4
a3 + a5 = 1 + (−5) = −4
a4 + a4 = (−2) + (−2) = −4
Notice each of these sums is equal to the sum of the first and last terms, and that we added
the center term, a4 , to itself. (Had we chosen an even number of terms, there would have
been no center term, and this would not have been necessary.)
So the sum (a1 + a2 + · · · + a7 ) + (a7 + a6 + · · · + a1 ) = 7(−4) = 7(a1 + a7 ). In other
words, 2(a1 + a2 + · · · + a7 ) = 7(a1 + a7 ). so a1 + a2 + · · · + a7 = 7(a12+a7 ) , which is of the form
(number of terms)[(first term)+(last term)]
. In fact, this pattern holds for every arithmetic
2
sequence.
Theorem 1.1. If a1 , a2 , a3 , . . . , an is an arithmetic sequence, then
a1 + a2 + a3 + · · · + an =
n(a1 + an )
.
2
Proof. Example 1.6 is our guide. If we can show that the nested sums a1 + an , a2 + an−1 ,
a3 + an−2 , etc., are all equal, then it will follow that
(a1 + a2 + a3 + · · · + an ) + (an + an−1 + an−2 + · · · + a1 ) = n(a1 + an ),
from which our result follows. The sum of each pair of numbers we add is of the form
a1+i + an−i for some i between 0 and n − 1. Using the fact that ak = a + (k − 1)d, we have
a1+i + an−i = (a + ([1 + i] − 1)d) + (a + ([n − i] − 1)d)
= (a + (i)d) + (a + (n − i − 1)d)
= (a) + (a + (n − i − 1)d + (i)d)
= (a) + (a + (n − 1)d)
= a1 + an .
Since each of these nested pairs sum to a1 + an , we have
2(a1 + a2 + · · · + an−1 + an ) = (a1 + a2 + · · · + an−1 + an ) + (an + an−1 + · · · + a2 + a1 )
= (a1 + an ) + (a2 + an−1 ) + · · · + (an−1 + a2 ) + (an + a1 )
!
"#
$
n terms
= (a1 + an ) + (a1 + an ) + · · · + (a1 + an ) + (a1 + an )
!
"#
$
n terms
= n(a1 + an ).
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VAN WYK’S MATH 304
Thus we have our result.
!
Turning to the sum of a finite number of terms of a geometric sequence, use of the distributive law of multiplication over addition makes the derivation easier than that of the
sum of the terms of an arithmetic sequence. It also results in a different form for the sum.
Theorem 1.2. If a, ar, ar2 , . . . , ar n−1 is a geometric sequence, then
a + ar + ar 2 + · · · + ar n−1 =
a(r n − 1)
.
r−1
Proof. We’ll denote the sum of the first n terms of the geometric series as sn , so sn =
a + ar + ar 2 + · · · + ar n−1 . The technique we will use is similar to one you may have used
when calculating the fractional form of a number with a repeating decimal representation;
we will multiply both sides of the above equation by r and then subtract them to eliminate
all but two terms:
rsn − sn = r(a + ar + ar 2 + · · · + ar n−1 ) − (a + ar + ar 2 + · · · + ar n−1 )
= (ar + ar 2 + ar 3 + · · · + ar n ) − (a + ar + ar 2 + · · · + ar n−1 )
= ar n − a.
But then (r − 1)sn = a(r n − 1), so sn =
a(r n −1)
,
r−1
as desired.
!
n
n
n
−1)
−1)
)
In the expression a(rr−1
, n is the number of terms in the sum. Notice that a(rr−1
= a(1−r
;
1−r
you could use the former if r > 1 and the latter if r < 1, although either will obviously give
the same result.
Example 1.7. You have seen examples of sums of geometric sequences in the form of finite
33333
repeating decimals. The rational number 100000
has decimal representation 0.33333, which is
3
and common ratio
the sum of the first five terms of the geometric series with initial term 10
1
:
10
% &2
% &3
% &4
3 1
3
1
3
1
3
1
3
+
·
+
·
·
·
+
+
.
0.33333 =
10 10 10 10
10
10
10
10
10
More remarkable examples occur with infinite repeating decimals, such as
1
3
= 0.333. In
1
and common ratio 10
, but
this case, we again have a geometric series with initial term
the simple expression 0.333 actually represents the sum of infinitely many positive numbers,
' 1 (n
3
· 10
. The study of infinite sums of series is one of the topics of Calculus.
each of the form 10
3
10
In summary, here is what we have for arithmetic and geometric sequences:
FUNDAMENTALS OF ALGEBRA
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Type of Sequence
Form
nth term
Sum of 1st n Terms
Arithmetic
a, a + d, a + 2d, . . .
a + (n − 1)d
n(a1 +an )
2
Geometric
a, ar, ar2 , . . .
ar n−1
a(r n −1)
r−1
Problems.
1. Give an example of each of the following, if possible:
(a) A real number that is not an integer.
(b) A rational number that is not an integer.
(c) A nonpositive integer.
(d) A number that is both an integer and a rational number.
(e) A rational number that is not a real number.
2. Give a recursive definition of the set of positive integers that are multiples of 7, that is,
the set {7, 14, 21, 28, . . . }.
3. Triangular numbers have the following recursive definition:
Base Step: t1 = 1.
Recursive Step: tn = tn−1 + n (for n ≥ 2).
Compute the first six triangular numbers and draw dot pictures4 to show why they are
called triangular numbers.
4. Find the 451st term of the arithmetic sequence 3, 3 14 , 3 21 , 3 34 , 4, 4 41 , . . . .
27 81 243
5. Find the 451st term of the geometric sequence 3, 94 , 16
, 64 , 256 , . . . .
6. The nth triangular number from Problem 3 is tn = 1 + 2 + 3 + · · · + n. Thinking of this
as a sum of the first n terms of an arithmetic sequence, use Theorem 1.1 to find a “closed
form5” for tn , and use it to compute t100 .
7. Jason got $100 from his uncle for his tenth birthday and put it under his mattress for
safekeeping. Each birthday after that, Jason put an additional $60 under his mattress,
and never took any money out. On which birthday will Jason first be able to afford a
$700 mountain bike, assuming tax is including in that price?
8. Olivia’s starting salary was $50,000, and she was guaranteed an increase in salary of 5%
on the first day of every year after that. What was Olivia’s salary on the first day of her
seventh year of employment?
4Draw
5In
1 dot (•) to represent the number 1, 3 dots to represent the number 3, etc.
this case, a closed form is one that does not contain any elipses (. . . ).
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VAN WYK’S MATH 304
9. Adam dropped a ball from 100 feet in the air. Every time it hit the ground, it bounced
two-thirds as high as it fell. What was the total distance the ball traveled the instant it
hit the ground the seventh time?
10. The isotope nickel-63 has a half-life of approximately 100 years6. Amy the geologist buried
50 kg of nickel-63 in a time capsule on January 1, 2000. How much nickel-63 will be in
the capsule on January 1, 3000?
11. Square numbers are positive integers that are perfect squares: 1, 4, 9, 16, 25, 36, 49, . . . .
(a) Draw dot pictures to show one reason these are called square numbers.
(b) Use these pictures to show geometrically that n2 = tn + tn−1 , where tn is the nth
triangular number from Problem 3.
Hint. Start with a dot picture for n2 and try to subdivide it into two triangles.
(c) Use the closed-form for tn in Problem 6 to show algebraically that n2 = tn + tn−1 .
12. Here is an old puzzle:
An adult and two children had to cross a river by boat. The boat could only
hold one adult or at most two children. How can the adult and two children all
get to the other side?
(a) Answer the puzzle, and count how many one-way trips must be made. (Remember:
someone always has to take the boat across.)
(b) Now assume there are n adults and two children. How many one-way trips must be
made to get all n + 2 people to the other side?
6In
other words, half of it decays every one hundred years.