nth Terms of Linear Sequences (Cambridge International IGCSE Maths)

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 = 2*n * + 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 = -5*n * + 20

Worked Example Find a formula for the n^th term of the sequence -7, -3, 1, 5, 9, ...

Answer:

The n th term is dn + b where d is the common difference and b 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, ...

** b *= -11

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

Final answer: n^th term = 4n - 11