Process
What is a Sequence?
A Sequence is a set
of things (usually numbers) that are in order. 
Infinite or Finite
If the sequence goes
on forever it is called an infinite sequence,
otherwise it is a finite sequence
otherwise it is a finite sequence
Examples:
{1, 2, 3, 4 ,...} is a very simple sequence (and it is
an infinite sequence)
{20, 25, 30, 35, ...} is also an infinite sequence
{1, 3, 5, 7} is the sequence of the first 4 odd
numbers (and is a finite sequence)
{4, 3, 2, 1} is 4 to 1 backwards
{1, 2, 4, 8, 16, 32, ...} is an infinite sequence
where every term doubles
{a, b, c, d, e} is the sequence of the first 5 letters
alphabetically
{f, r, e, d} is the sequence of letters in the name
"fred"
{0, 1, 0, 1, 0, 1, ...} is the sequence of alternating
0s and 1s (yes they are in order, it is an alternating order in this case)
In Order
When we say the terms are "in order", we are
free to define what order that is! They could go forwards, backwards ...
or they could alternate ... or any type of order you want!
Like a Set
- the terms are in order (with Sets the order does not matter)
- the same value can appear many times (only once in Sets)
Example: {0, 1, 0, 1, 0, 1, ...} is the sequence
of alternating 0s and 1s.
The set would be just {0,1}
A Rule
A Sequence usually has a Rule, which is a way
to find the value of each term.
Example: the sequence {3, 5, 7, 9, ...} starts at 3
and jumps 2 every time:
As a Formula
Saying "starts at 3 and jumps 2 every time"
is fine, but it doesn't help us calculate the:
- 10th term,
- 100th term, or
- nth term, where n could be any term number we want.
So, we want a formula
with "n" in it (where n is any term number).
So, What Would A Rule
For {3, 5, 7, 9, ...} Be?
Firstly, we can see the sequence goes up 2 every time,
so we can guess that a Rule will be something like "2 times n"
(where "n" is the term number). Let's test it out:
Test Rule: 2n
n
|
Term
|
Test Rule
|
1
|
3
|
2n = 2×1
= 2
|
2
|
5
|
2n = 2×2
= 4
|
3
|
7
|
2n = 2×3
= 6
|
That nearly worked ... but it is too low
by 1 every time, so let us try changing it to:
Test Rule: 2n+1
n
|
Term
|
Test Rule
|
1
|
3
|
2n+1 = 2×1
+ 1 = 3
|
2
|
5
|
2n+1 = 2×2
+ 1 = 5
|
3
|
7
|
2n+1 = 2×3
+ 1 = 7
|
That Works!
So instead of saying "starts at 3 and jumps 2
every time" we write this:
2n+1
Now we can calculate, for example, the 100th term:
2 × 100 + 1 = 201
Many Rules
But mathematics is so powerful we can find more
than one Rule that works for any sequence.
Example: the sequence
{3, 5, 7, 9, ...}
We have just shown a Rule for {3, 5, 7, 9, ...} is: 2n+1
And so we get: {3, 5,
7, 9, 11, 13, ...}
But can we find another rule?
How about "odd numbers without a 1 in
them":
And we would get: {3,
5, 7, 9, 23, 25, ...}
A completely different sequence!
And we could find more rules that match {3, 5, 7,
9, ...}. Really we could.
So it is best to say "A Rule" rather then
"The Rule" (unless you know it is the right Rule).
Notation
To make it easier to use rules, we often use this
special style:
Arithmetic Sequences
In other words, you just add some value each time ...
on to infinity.
Example:
1, 4, 7, 10, 13, 16, 19, 22, 25, ...
|
This sequence has a
difference of 3 between each number.
Its Rule is xn = 3n-2
Its Rule is xn = 3n-2
In General you could write an
arithmetic sequence like this:
{a, a+d, a+2d, a+3d,
... }
where:
- a is the first term, and
- d is the difference between the terms (called the "common difference")
And you can make the rule by:
xn = a +
d(n-1)
(We use
"n-1" because d is not used in the 1st term).
Geometric Sequences
Example:
2, 4, 8, 16, 32, 64, 128, 256, ...
|
This sequence has a
factor of 2 between each number.
Its Rule is xn = 2n
Its Rule is xn = 2n
In General you could write a
geometric sequence like this:
{a, ar, ar2,
ar3, ... }
where:
- a is the first term, and
- r is the factor between the terms (called the "common ratio")
Note: r should not be 0.
- When r=0, you get the sequence {a,0,0,...} which is not geometric
And the rule is:
xn = ar(n-1)
(We use
"n-1" because ar0 is the 1st term)
1, 3, 6, 10, 15, 21, 28, 36, 45, ...
|
By adding another row of dots and counting all the
dots we can find the next number of the sequence:
But it is easier to use this Rule:
xn =
n(n+1)/2
Example:
- the 5th Triangular Number is x5 = 5(5+1)/2 = 15,
- and the sixth is x6 = 6(6+1)/2 = 21
Square Numbers
1, 4, 9, 16, 25, 36, 49, 64, 81, ...
|
The next number is made by squaring where it is in the
pattern.
Rule is xn
= n2
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