Introduction to Sequences (Edexcel GCSE Maths: Foundation)

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Introduction to Sequences

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 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
      • an represents the nth term

A sequence of numbers

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 = 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 8 + 2
    • Is 98 in the sequence?
      table row cell 8 n plus 2 end cell equals 98 row cell 8 n end cell equals 96 row n equals cell 96 over 8 end cell row n equals 12 end table
      • It is in the sequence, it is the 12th term
    • Is 124 in the sequence?
      table row cell 8 n plus 2 end cell equals 124 row cell 8 n end cell equals 122 row n equals cell 122 over 8 end cell row n equals cell 15.25 end cell end table
      • n  is not a whole number, so it is not in the sequence

Examiner Tip

  • In the exam, it helps to write the position number (the value of n) above each term in the sequence.

Worked example

A sequence has the n th term formula 3n  + 2.

(a)
Find the first three terms in the sequence.
 

Substitute n  = 1, = 2 and n  = 3 into the formula

3×1 + 2 = 5

3×2 + 2 = 8

3×3 + 2 = 11

5, 8, 11

 
(b)
Find the 80th term.
  

Substitute  = 80 into the formula

3×80 + 2

The 80th term is 242

  
(c)
Determine whether the number 96 is in the sequence.
  

Set the formula equal to 96

3 n plus 2 equals 96

Solve to find n
If n is a whole number, it is a term in the sequence

table attributes columnalign right center left columnspacing 0px end attributes row cell 3 n end cell equals cell 96 minus 2 end cell row cell 3 n end cell equals 94 row n equals cell 94 over 3 end cell end table

94 is not divisible by 3
The nearest possible value is 95 ((95-2) ÷ 3 = 31, the 31st term)

96 is not in the sequence

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Mark

Author: Mark

Expertise: Maths

Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.