Linear Sequences (Edexcel GCSE Maths: Foundation)

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Linear Sequences

What is a linear sequence?

  • A linear sequence goes up (or down) by the same amount each time
  • This amount is called the common difference, d 
    • For example:
      1, 4, 7, 10, 13, …(adding 3, so d = 3)
      15, 10, 5, 0, -5, … (subtracting 5, so d = -5)
  • Linear sequences are also called arithmetic sequences
  • You can form a linear sequence by using the numbers in a times table 
    • 3n  is the sequence of numbers in the 3 times table
      • 3, 6, 9, 12, 15, ...
    • 3n + 1  is the sequence of numbers that are 1 more than the numbers in the 3 times table
      • 4, 7, 10, 13, 16, ...

How do I find the nth term formula for a linear sequence?

  • The formula is n th term = dn  + b
    • is the common difference
      • The amount it goes up by each time
    • is the value before the first term (sometimes called the zero term)
      • Imagine going backwards
  • For example 5, 7, 9, 11, ....
    • The sequence adds 2 each time
      • d  = 2
    • Now continue the sequence backwards, from 5, by one term
      • (3), 5, 7, 9, 11, ...
      • b  = 3
    • So the n th term = 2 + 3
  • For example 15, 10, 5, ...
    • Subtracting 5 each time means d  = -5
    • Going backwards from 15 gives 15 + 5 = 20
      • (20), 15, 10, 5, ... so = 20
    • The n th term = -5 + 20

Worked example

Find a formula for the nth term of the sequence -7, -3, 1, 5, 9, ...

The n th term is dn  + where is the common difference and is the term before the 1st term
The sequence goes up by 4 each time

d  = 4

Continue the sequence backwards by one term (-7-4) to find b

(-11), -7, -3, 1, 5, 9, ...

= -11

Substitute = 4 and b  = -11 into dn  + b

nth term = 4n  - 11

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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.