nth Terms of Linear Sequences (AQA GCSE Maths): Revision Note

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

How do I find the n^th term formula for a linear sequence?

• The formula is n th term = dn  + b

  • d  is the common difference

    • The amount it goes up by each time

  • b  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 = 2n  + 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 b  = 20

  • The n th term = -5n  + 20