SEQUENCES AND NUMBER PATTERN

Sequences

You can read a gentle introduction to Sequences in Common Number Patterns.

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).

( http://www.mathsisfun.com/algebra/sequences-series.html )