Introduction to Sequences (Edexcel IGCSE Maths)

What are sequences?

• A sequence is an ordered set of (usually) numbers
• 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 often used for (an unknown) position
  • For the first term, n = 1 and for the second term, n = 2, and so on

• Subscript notation is used to talk about a particular term
  • a₁ would be the first term in a sequence
  • a₇ would be the seventh term
  • a_n would be the n^th term

What is a position-to-term rule?

• A position-to-term rule gives the n^th term of a sequence in terms of n
  • This is a very powerful piece of mathematics
  • With a position-to-term rule the 100^th term of a sequence can be found without having to know or work out the first 99 terms!

What is a term-to-term rule?

• A term-to-term rule gives the (n+1)^th term in terms of the n^th term
  • ie a_n+1 is given in terms of a_n
  • If a term is known, the next one can be worked out

How do I use a position-to-term rule to write the first n terms of a sequence? 

• To generate the first n terms of a sequence using a position-to-term rule (an n^th term rule), substitute n = 1, n = 2, n = 3, and so on, into the rule

• For example, using a_n = 2n + 3 to generate the first four terms;
  • 1st term: n = 1 so a₁ = 2(1) + 3 = 5
  • 2nd term: n = 2 so a₂ = 2(2) + 3 = 7
  • 3rd term: n = 3 so a₃ = 2(3) + 3 = 9
  • 4th term: n = 4 so a₄ = 2(4) + 3 = 11
  • Sequence is 5, 7, 9, 11, ...

• Another example, using a_n = n² − 5 to generate the first four terms;
  • a₁ = 1² − 5 = -4
  • a₂ = 2² − 5 = -1
  • a₃ = 3² − 5 = 4
  • a₄ = 4² − 5 = 11
  • Sequence is -4, -1, 4, 11, ... (This is an example of a quadratic sequence)

How do I use a term-to-term rule to write the first n terms of a sequence? 

• To generate the first n terms of a sequence using a term-to-term rule (an n^th term rule), you need to be given the first term (a₁) and the term-to-term rule
• The term-to-term rule may be given in the form "a_n+1 = ... " where a_n+1 means the next term

• For example, a₁ = 5 and a_n+1 = a_n + 2, generate the first four terms;
  • 1st term, a₁ = 5
  • 2nd term, a₂ = a₁ + 2 = 5 + 2 = 7
  • 3rd term, a₃ = a₂ + 2 = 7 + 2 = 9
  • 4th term, a₄ = a₃ + 2 = 9 + 2 = 11
  • Sequence is 5, 7, 9, 11, ... Notice that this is the same sequence that was generated above using the position-to-term rule a_n = 2n + 3

• Another example, a₁ = 5 and a_n+1 = 2a_n, generate the first four terms;
  • a₁ = 5
  • a₂ = 2a₁ = 2 × 5 = 10
  • a₃ = 2a₂ = 2 × 10 = 20
  • a₄ = 2a₄ = 2 × 20 = 40
  • Sequence is 5, 10, 20, 40, ... (This is an example of a geometric sequence)

• A final example, a₁ = 1, a₂ = 1 and a_n+2 = a_n+1 + a_n, generate the first four terms;
  • • a₁ = 1 and a₂ = 1
    • a₃ = a₂ + a₁ = 1 + 1 = 2
    • a₄ = a₃ + a₂ = 2 + 1 = 3
    • a₅ = a₄ + a₃ = 3 + 2 = 5
    • Sequence is 1, 1, 2, 3, 5, ... (This is the Fibonacci sequence) 

What types of sequences are there?

• You may have heard of special types sequences like geometric sequences, quadratic sequences or Fibonacci sequences. It's good to know that they exist and as we've seen above, you may be asked to generate them using position-to-term and term-to-term rules
• However for your Edexcel IGCSE exams, you do not need to have any specific knowledge about these types of sequence
• The only specific type of sequence you need to know about for Edexcel IGCSE exams are arithmetic sequences, which are explained in the next section