nth Terms of Linear Sequences (Cambridge (CIE) IGCSE International Maths)
Revision Note
Written by: Mark Curtis
Reviewed by: Dan Finlay
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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
How do I find the nth 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
Worked Example
Find a formula for the nth term of the sequence -7, -3, 1, 5, 9, ...
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
nth term = 4n - 11
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