Quadratic & Cubic Sequences (Cambridge (CIE) IGCSE International Maths)
Revision Note
Written by: Mark Curtis
Reviewed by: Dan Finlay
Quadratic & Cubic Sequences
What is a quadratic sequence?
A quadratic sequence has an n th term formula that involves n2
The second differences are constant (the same)
These are the differences between the first differences
For example, 3, 9, 19, 33, 51, …
1st Differences: 6, 10, 14, 18, ...2nd Differences: 4, 4, 4, ...
How do I find the nth term formula for a simple quadratic sequence?
The sequence with n th term formula n2 are the square numbers
1, 4, 9, 16, 25, 36, 49, ...
From 12, 22, 32, 42, ...
Finding the n th term formula comes from comparing sequences to the square numbers
2, 5, 10, 17, 26, 37, 50, ... has the formula n2 + 1
Each term is 1 more than the square numbers
2, 8, 18, 32, 50, 72, 98, ... has the formula 2n2
Each term is double a square number
It may be a simple combination of both
For example, doubling then adding 1
3, 9, 19, 33, 51, 73, 99, ... has the formula 2n2 + 1
Some sequences are just the square numbers but starting later
16, 25, 36, 49, ... has the formula (n + 3)2
Substitute in n = 1, n = 2, n = 3 to see why
What is a cubic sequence?
A cubic sequence has an n th term formula that involves n3
The third differences are constant (the same)
These are the differences between the second differences
For example, 4, 25, 82, 193, 376, 649, ...
1st Differences: 21, 57, 111, 183, 273, ...2nd Differences: 36, 54, 72, 90, ...
3rd Differences: 18, 18, 18, ...
How do I find the nth term formula for a simple cubic sequence?
The sequence with n th term formula n3 are the cube numbers
1, 8, 27, 64, 125, ...
From 13, 23, 33, 43, ...
Finding the n th term formula comes from comparing sequences to the cube numbers
2, 9, 28, 65, 126, ... has the formula n3 + 1
Each term is 1 more than the cube numbers
2, 16, 54, 128, 250, ... has the formula 2n3
Each term is double a cube number
Can I use the differences to help find the nth term formula?
Yes, for the simple quadratic sequence an2 + b
a is
of the second difference
For the simple cubic sequence an3 + b
a is
of the third difference
Examiner Tips and Tricks
You must learn the square numbers from 12 to 152 and the cube numbers from 13 to 53 (and know 103)
Worked Example
For the sequence 6, 9, 14, 21, 30, ....
(a) Find a formula for the nth term.
Find the first and second differences
Sequence: 6, 9, 14, 21, 30
First differences: 3, 5, 7, 9, ...
Second differences: 2, 2, 2, ...
The second differences are constant so this must be a quadratic sequence
Compare each term to terms in the sequence n2 (the square numbers)
n2 : 1, 4, 9, 16, 25, ...
Sequence: 6, 9, 14, 21, 30, ...
Each term is 5 more than the terms in n2, so add 5 to n2
nth term = n2 + 5
You could also have used an2 + b where a is half the second difference
(b) Hence, find the 20th term of the sequence.
Substitute n = 20 into n2 + 5
(20)2 + 5 = 400 + 5
The 20th term is 405
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