Quadratic Sequences
Neil Trivedi
Teacher
Contents
Quadratic Sequences
Previously, we covered linear sequences, which have an term of the form In this note, we cover quadratic sequences, which have an term of the form
where and are constants and
We will use the first few terms of a quadratic sequence to find its term, which is what we usually mean by “finding the rule”.
To begin, we need to derive a set of rules that link the constants and to the terms of the sequence.
We do this by substituting and into to get the first four terms in terms of and
Next, we find the differences between consecutive terms. These are called the first differences.
Then, we find the differences between these first differences. These are called the second differences.
Notice that the second difference is always This means that, no matter how far along the sequence we go, the second difference stays the same: it is constant.
Key idea: the constant second difference is the defining property of a quadratic sequence. It is also how we tell whether a given sequence is quadratic in the first place. If the second difference is constant, the sequence is quadratic.
From these results, we can pick out three useful rules that link and to features of the sequence:
Second difference
First difference between term and term
First term
We use these three equations to find and in turn, and then substitute them back into to write down the term.
For a quadratic sequence in the form we can find the constants and by following these three rules in order:
Second difference
First difference between term and term
First term
Example 1:
Find the term of the following quadratic sequence:
Step 1: Find the first and second differences.
The second difference is constant which confirms the sequence is quadratic.
Step 2: Use the three rules to find the values of and
Using the rule which states that second difference:
Dividing both sides by
Using the rule which states that first difference between term and term
Substituting
Subtracting from both sides,
Using the rule which states that first term:
Substituting and
Subtracting from both sides,
Step 3: Substitute these values into
term
Example 2:
The first five terms of a quadratic sequence are
Determine whether is a term of this sequence. You must show your working.
Idea: To check if is a term, we first find the term of the sequence. We then set the term equal to and solve for If comes out as a positive integer, then is a term. This is because represents a term number, which must be a counting number. Otherwise, is not a term.
Step 1: Find the first and second differences.
Step 2: Use the three rules to find and
Then,
Finally,
So, the term is
Step 3: Set the term equal to and solve for
We want to solve a positive quadratic so we will move all terms on the left-hand side to the right by adding and subtracting both and from both sides to set one side equal to
We now factorise this quadratic using the OI-OI method (see our Factorising Quadratics note if you need a refresher). After looking for factors of whose outer-inner products combine to give we obtain:
Setting each bracket equal to
or
or
Since must be a positive integer, we reject So,
Therefore, is a term of the sequence, and it is the term.
Example 3:
The first four patterns of a sequence of dots are shown below.
a) Find an expression, in terms of for the number of dots in the pattern.
Step 1: Count the dots in each pattern to form a sequence.
By counting the dots in each pattern, we obtain the sequence:
This is a sequence of numbers, just like in the previous examples. We can now use the same method to find the term.
Step 2: Find the first and second differences.
Step 3: Use the three rules to find and
Then,
Finally,
So, the term is
b) Hence, find the number of dots in the pattern.
To find the number of dots in the pattern, we substitute into the expression found in part a).
term
Therefore, the pattern contains dots.
Practice Question