Sequences (CIE IGCSE Maths: Core)

What are sequences?

• A sequence is an ordered set of numbers that follow a rule
  • For example 3, 6, 9, 12...
    • The rule is to add 3 each time
• Each number in a sequence is called a term
  

• The location of a term within a sequence is called its position
  • The letter n  is used for position
    • n  = 1 refers to the 1st term
    • n  = 2 refers to the 2nd term
    • If you do not know its position, you can say the n th term
• Another way to show the position of a term is using subscripts
  • A general sequence is given by a₁, a₂, a₃, ...
    
    • a₁ represents the 1st term
    • a₂ represents the 2nd term
    • a_n represents the nth term

How do I write out a sequence using a term-to-term rule?

• Term-to-term rules tell you how to get the next term from the term you are on
  • It is what you do each time
  • For example, starting on 4, add 10 each time
    • 4, 14, 24, 34, ...

How do I write out a sequence using a position-to-term rule?

• A position-to-term rule is an algebraic expression in n  that lets you find any term in the sequence
  • This is also called the n th term formula
• You need to know what position in the sequence you are looking for
  • • To get the 1st term, substitute in n  = 1
    • To get the 2nd term, substitute in n  = 2
• You can jump straight to the 100th term by substituting in n  = 100
  • You do not need to find all 99 previous terms
• For example, the n th term is 8n  + 2
  • The 1st term is 8×1 + 2 = 10
  • The 2nd term is 8×2 + 2 = 18
  • The 100th term is 8×100 + 2 = 802

How do I know if a value belongs to a sequence?

• If you know the n th term formula, set the value equal to the formula
  • This creates an equation to solve for n
• For example, a sequence has the n th term formula 8n  + 2
  • Is 98 in the sequence?
    
    
    • It is in the sequence, it is the 12th term
  • Is 124 in the sequence?
    
    • n  is not a whole number, so it is not in the sequence

How do I continue a given sequence? 

• You can work out the first differences to see if there is a pattern
  • The first differences are the values the sequence changes by each time
• For example
  • 
    • The first differences are all +3
    • The next term is 13 + 3 = 16
  • 
    • The first differences are all -5
    • The next term is -9 - 5 = -14 
  • 
    • The first differences increase by 1
    • The next term is 23 + 9 = 32
  •  
    • The first differences double each time
    • The next term is 15 + 16 = 31

Sequences of squares, cubes, and triangular numbers

• Sequences can often be formed using square, cube, or triangular numbers
• It can help to be familiar with these sequences of numbers
• Square numbers are the results of squaring integers
  • 1²,  2²,  3²,  4²,  5²,  6²,  7²,  8²,  9²,  10²,  11²,  12²,  ...
  • 1,  4,  9,  16,  25,  36,  49,  64,  81,  100,  121,  144,  ...
• Cube numbers are the results of cubing integers
  • 1³,  2³,  3³,  4³,  5³,  ...
  • 1,  8,  27,  64,  125,  ...
• Triangular numbers are the result of summing consecutive integers
  
  • 1,  1+2,  1+2+3,  1+2+3+4,  1+2+3+4+5,  ...
  • 1,  3,  6,  10,  15, ...
  • When drawn as dots, triangular numbers form triangles
    •