In Grade 10, you learned about arithmetic sequences, where the difference between consecutive terms was constant. In this chapter we learn about quadratic sequences.
What is a
quadratic sequence ?
Quadratic Sequence
A quadratic sequence is a sequence of numbers in which the second differences between each consecutive term differ by the same amount, called a common second difference.
For example,
is a quadratic sequence. Let us see why ...
If we take the difference between consecutive terms, then:
We then work out the
second differences , which is simply obtained by taking the difference between the consecutive differences {
} obtained above:
We then see that the second differences are equal to 1. Thus,
[link] is a
quadratic sequence .
Note that the differences between consecutive terms (that is, the first differences) of a quadratic sequence form a sequence where there is a constant difference between consecutive terms. In the above example, the sequence of {
}, which is formed by taking the differences between consecutive terms of
[link] , has a linear formula of the kind
.
Quadratic sequences
The following are also examples of quadratic
sequences:
Can you calculate the common second difference for each of the above examples?
Write down the next two terms and find a formula for the
term of the sequence
i.e.
the second difference is 4.
So continuing the sequence, the differences between each term will be:
So the next two terms in the sequence willl be:
So the sequence will be:
We know that the second difference is 4. The start of the formula will therefore be
.
If
, you have to get the value of term one, which is 5 in this particular sequence. The difference between
and original number (5) is 3, which leads to
.
Check if it works for the second term, i.e. when
.
Then
. The difference between term two (12) and 8 is 4, which is can be written as
.
So for the sequence
the formula for the
term is
.
General case
If the sequence is quadratic, the
term should be
TERMS
difference
difference
In each case, the second difference is
.
This fact can be used to find
, then
then
.
The following sequence is quadratic:
Find the rule.
TERMS
8
22
42
68
difference
14
20
26
difference
6
6
6
The rule is therefore:
For
Derivation of the
-term of a quadratic sequence
Let the
-term for a quadratic sequence be given by