Sequences 

What is a Sequence?

A Sequence is a list of things (usually numbers) that are in order.

Infinite or Finite

When the sequence goes on forever it is called an infinite 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 we want!

Like a Set

A Sequence is like a Set, except:

• 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 is just {0,1}

Notation

Sequences also use the same notation as sets: list each element, separated by a comma, and then put curly brackets around the whole thing.

{3, 5, 7, ...}

The curly brackets { } are sometimes called "set brackets" or "braces".

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:

• 10^th term,
• 100^th term, or
• n^th 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 Can 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 is 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 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 we know it is the right Rule)

Notation

To make it easier to use rules, we often use this special style:

xn is the term n is the term number

Example: to mention the "5th term" we write: x₅

So a rule for {3, 5, 7, 9, ...} can be written as an equation like this:

x_n = 2n+1

And to calculate the 10th term we can write:

x₁₀ = 2n+1 = 2×10+1 = 21

Can you calculate x₅₀ (the 50th term) doing this?

Here is another example:

Example: Calculate the first 4 terms of this sequence:

{a_n} = { (-1/n)ⁿ }

Calculations:

• a₁ = (-1/1)¹ = -1
• a₂ = (-1/2)² = 1/4
• a₃ = (-1/3)³ = -1/27
• a₄ = (-1/4)⁴ = 1/256

Answer:

{a_n} = { -1, 1/4, -1/27, 1/256, ... }

Special Sequences

Now let's look at some special sequences, and their rules.

Arithmetic Sequences

In an Arithmetic Sequence the difference between one term and the next is a constant.
In other words, we 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 x_n = 3n-2

In General we can 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 we can make the rule:

x_n = a + d(n-1)

(We use "n-1" because d is not used in the 1st term).

Geometric Sequences

Geometric Sequences and Sums

Sequence

A Sequence is a set of things (usually numbers) that are in order.

Geometric Sequences

In a Geometric Sequence each term is found by multiplying the previous term by a constant.

Example:

1, 2, 4, 8, 16, 32, 64, 128, 256, ...

This sequence has a factor of 2 between each number.

Each term (except the first term) is found by multiplying the previous term by 2.

In General we 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")

Example: {1,2,4,8,...}

The sequence starts at 1 and doubles each time, so

• a=1 (the first term)
• r=2 (the "common ratio" between terms is a doubling)

And we get:

{a, ar, ar², ar³, ... }

= {1, 1×2, 1×2², 1×2³, ... }

= {1, 2, 4, 8, ... }

But be careful, r should not be 0:

• When r=0, we get the sequence {a,0,0,...} which is not geometric

The Rule

We can also calculate any term using the Rule:

x_n = ar^(n-1)

(We use "n-1" because ar⁰ is for the 1st term)

Example:

10, 30, 90, 270, 810, 2430, ...

This sequence has a factor of 3 between each number.

The values of a and r are:

• a = 10 (the first term)
• r = 3 (the "common ratio")

The Rule for any term is:

x_n = 10 × 3^(n-1)

So, the 4th term is:

x₄ = 10×3^(4-1) = 10×3³ = 10×27 = 270

And the 10th term is:

x₁₀ = 10×3^(10-1) = 10×3⁹ = 10×19683 = 196830

A Geometric Sequence can also have smaller and smaller values:

Example:

4, 2, 1, 0.5, 0.25, ...

This sequence has a factor of 0.5 (a half) between each number.

Its Rule is x_n = 4 × (0.5)^n-1

Why "Geometric" Sequence?

Because it is like increasing the dimensions in geometry:

	a line is 1-dimensional and has a length of r
	in 2 dimensions a square has an area of r2
	in 3 dimensions a cube has volume r3
	etc (yes we can have 4 and more dimensions in mathematics).

Geometric Sequences are sometimes called Geometric Progressions (G.P.’s)

Summing a Geometric Series

When we need to sum a Geometric Sequence, there is a handy formula.
To sum:

a + ar + ar² + ... + ar^(n-1)

Each term is ar^k, where k starts at 0 and goes up to n-1

Use this formula:

a is the first term
r is the "common ratio" between terms
n is the number of terms

What is that funny Σ symbol? It is called Sigma Notation

(called Sigma) means "sum up"

And below and above it are shown the starting and ending values:

It says "Sum up n where n goes from 1 to 4. Answer=10

The formula is easy to use ... just "plug in" the values of a, r and n

Example: Sum the first 4 terms of

10, 30, 90, 270, 810, 2430, ...

This sequence has a factor of 3 between each number.

The values of a, r and n are:

• a = 10 (the first term)
• r = 3 (the "common ratio")
• n = 4 (we want to sum the first 4 terms)

So:

Becomes:

You can check it yourself:

10 + 30 + 90 + 270 = 400

And, yes, it is easier to just add them in this example, as there are only 4 terms. But imagine adding 50 terms ... then the formula is much easier.

Using the Formula

Let's see the formula in action:

Example: Grains of Rice on a Chess Board

On the page Binary Digitswe give an example of grains of rice on a chess board. The question is asked:
When we place rice on a chess board:

• 1 grain on the first square,
• 2 grains on the second square,
• 4 grains on the third and so on,
• ...

... doubling the grains of rice on each square ...

... how many grains of rice in total?

So we have:

• a = 1 (the first term)
• r = 2 (doubles each time)
• n = 64 (64 squares on a chess board)

So:

Becomes:

= 1−2⁶⁴−1 = 2⁶⁴ − 1

= 18,446,744,073,709,551,615

Which was exactly the result we got on the Binary Digits page (thank goodness!)

And another example, this time with r less than 1:

Example: Add up the first 10 terms of the Geometric Sequence that halves each time:

{ 1/2, 1/4, 1/8, 1/16, ... }

The values of a, r and n are:

• a = ½ (the first term)
• r = ½ (halves each time)
• n = 10 (10 terms to add)

So:

Becomes:

Very close to 1.

(Question: if we continue to increase n, what happens?)

Why Does the Formula Work?

Let's see why the formula works, because we get to use an interesting "trick" which is worth knowing.

First, call the whole sum "S":  S = a + ar + ar² + ... + ar⁽ⁿ⁻²⁾+ ar⁽ⁿ⁻¹⁾

Next, multiply S by r:S·r = ar + ar² + ar³ + ... + ar⁽ⁿ⁻¹⁾ + arⁿ

Notice that S and S·r are similar?

Now subtract them!

Wow! All the terms in the middle neatly cancel out. 
(Which is a neat trick)

By subtracting S·r from S we get a simple result:

S − S·r = a − arⁿ

Let's rearrange it to find S:

Factor out S and a:S(1−r) = a(1−rⁿ)

Divide by (1−r):S = a(1−rⁿ)(1−r)

Which is our formula (ta-da!):

Infinite Geometric Series

So what happens when n goes to infinity?

Well ... when r is less than 1, then rⁿ goes to zeroand we get:

NOTE: this does not work when r is 1 or more (or less than -1):

r must be between (but not including) -1 and 1

and r should not be 0 because we get the sequence {a,0,0,...} which isn't geometric

Let's bring back our previous example, and see what happens:

Example: Add up ALL the terms of the Geometric Sequence that halves each time:

{ 12, 14, 18, 116, ... }

We have:

• a = ½ (the first term)
• r = ½ (halves each time)

And so:

= ½×1½ = 1

Yes, adding 12 + 14 + 18 + ... etc equals exactly 1.

Don't believe me? Just look at this square: By adding up 12 + 14 + 18 + ... we end up with the whole thing!

Recurring Decimal

On another page we asked "Does 0.999... equal 1?", well, let us see if we can calculate it:

Example: Calculate 0.999...

We can write a recurring decimal as a sum like this:

And now we can use the formula:

Yes! 0.999... does equal 1.

Example:

2, 4, 8, 16, 32, 64, 128, 256, ...

This sequence has a factor of 2 between each number.
Its Rule is x_n = 2ⁿ

In General we can 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, we get the sequence {a,0,0,...} which is not geometric

And the rule is:

x_n = ar^(n-1)

(We use "n-1" because ar⁰ is the 1st term)

Triangular Numbers

1, 3, 6, 10, 15, 21, 28, 36, 45, ...

The Triangular Number Sequence is generated from a pattern of dots which form a triangle:

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:

x_n = 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 x_n = n²

Cube Numbers

1, 8, 27, 64, 125, 216, 343, 512, 729, ...

The next number is made by cubing where it is in the pattern.

Rule is x_n = n³

Fibonacci Sequence

This is the Fibonacci Sequence

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

The next number is found by adding the two numbers before it together:

• The 2 is found by adding the two numbers before it (1+1)
• The 21 is found by adding the two numbers before it (8+13)
• etc...

Rule is x_n = x_n-1 + x_n-2

That rule is interesting because it depends on the values of the previous two terms.

Rules like that are called recursive formulas.

The Fibonacci Sequence is numbered from 0 onwards like this:

n =	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	...
xn=	0	1	1	2	3	5	8	13	21	34	55	89	144	233	377	...

Example: term "6" is calculated like this:

x₆ = x_6-1 + x_6-2 = x₅ + x₄ = 5 +