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)
- For example:
- 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, ...
- 3n is the sequence of numbers in the 3 times table
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
- d is the common difference
- 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
- The sequence adds 2 each time
- 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
