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Sequences and Series

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

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

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!

A Sequence is very much like a Set, but with the terms in order (and the same value can appear many times).

Example: {0, 1, 0, 1, 0, 1, ...} is the sequence of alternating 0s and 1s. The set would be just {0,1}

The Rule

A Sequence will have a Rule that gives you a way to find the value of each term.

Example: the sequence {3, 5, 7, 9, ...} starts at 3 and jumps 2 every time:

{3, 5, 7, 9, ...}

 

But The Rule Should be a Formula!

Saying "starts at 3 and jumps 2 every time" doesn't tell us how to 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 can be any term number).

So, What Would The Rule For {3, 5, 7, 9, ...} Be?

Firstly, we can see the sequence goes up 2 every time, so we can guess that the Rule will be something like "2 × 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 that Rule 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 the rule as

The Rule for {3, 5, 7, 9, ...} is: 2n+1

Now, for example, we can calculate the 100th term: 2 × 100 + 1 = 201

Notation

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

 

Term Number

It is common to use the style xn for terms:

  • xn is the term
  • n is the term number
  So to mention the "5th term" you just write: x5

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

xn = 2n+1

Now, if we want to calculate the 10th term we can write:

x10 = 2n+1 = 2×10+1 = 21

Can you calculate the 50th term? The 500th term?

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

Types of Sequences

Arithmetic Sequences

The example we just used {3,5,7,9,...} was an Arithmetic Sequence, because the difference between one term and the next is is a constant.

Examples

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


3, 8, 13, 18, 23, 28, 33, 38, ...
This sequence has a difference of 5 between each number.
Its Rule is xn = 5n-2

 

Geometric Sequences

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

Examples:

2, 4, 8, 16, 32, 64, 128, 256, ...
This sequence has a factor of 2 between each number.
Its Rule is xn = 2n

3, 9, 27, 81, 243, 729, 2187, ...
This sequence has a factor of 3 between each number.
Its Rule is xn = 3n

 

4, 2, 1, 0.5, 0.25, ...
This sequence has a factor of 0.5 (a half) between each number.
Its Rule is xn = 4 × 2-n

 

Special Sequences

Triangular Numbers

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

This 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.

 

triangular numbers

But it is easier to use the 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.

The Rule is xn = 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.

The Rule is xn = n3

 

Fibonacci Numbers

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...

The Rule is xn = xn-1 + xn-2

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

In the Fibonacci Numbers the terms are 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 would be calculated like this:

x6 = x6-1 + x6-2 = x5 + x4 = 5 + 3 = 8

Series

"Sequence" and "Series" might seem to be the same thing ... but in fact, a Series is the Sum of a Sequence.

Sequence: {1,2,3,4}

Series: 1+2+3+4 = 10

Series are often written down using Σ to mean "add them all up":

sum 1 to 4 This means "add up from 1 to 4" = 10
   
sum 2n+1 This means "add up the first four terms in the sequence 2n+1"

Which is the first four terms in our example sequence of {3,5,7,9,...} = 3+5+7+9 = 24