Quadratic and Cubic Sequences
Before reading this, make sure you are familiar with arithmetic sequences first!
(This is the method that most schools teach. There are multiple methods to accomplish problems of this category.)
Quadratic sequences go by such a form: an^2+bn+c
Unlike arithmetic sequences, quadratic sequences do not have a common difference, but still follow a kind of "pattern". Let me show you.
Here's one: 2n^2+n+0 (you can omit c since c=0 but I've included for clarity)
The first step is to obviously stick numbers into n to generate the sequence. I'm sticking consecutive intergers starting from 1.
3, 10, 21, 36, 55 Let's start by writing down the different terms in order
+7 +11 +15 +19 For the next step I've written down their differences below
+4 +4 +4
...And look! For the second difference (the difference between each term of the second row) is the same! Let's investigate this further:
Since all quadratic sequences follow the same form, let's try something else: subbing in numbers for n.
an^2+bn+c
n=1 n=2 n=3 n=4
a+b+c 4a+2b+c 9a+3b+c 16a+4b+c Sub n=x into the expression
3a+b 5a+b 7a+b What is the difference between each term? (Polynomial subtraction) 2a 2a The diference between the terms above this are all 2a.
As you can see, I've written down the differences between each term. And the first set of differences follow a particular pattern: there is a common difference between the first set of difference which is 2a.
What is this for? Well armed with this we can work out the expression of any quadratic sequence.
Why? Because those expressions is equal to the differences once you sub a and b into it.
Don't understand? Look at the first table.
a=2 b=1 c=0
The first term is 3. According to the expression that expresses the first term (topmost, highlighted in yellow), a+b+c should give us 3 once we fill in the correct a's,b's and c's.
2+1+0=3. Yep it works
Now look at the second column. 3a+b should give 7 because they are in the same position (so they represent the same things).
3(2)+(1)=7 Yep it works
Finally, look at the bottomost coner, the one with the second differences. (Named so because they are in the second row of differences between the terms)
2a=4
2(2)=4 Correct!
First work out all the differences like the 1st table.
To work out a, divide the numbers at the bottom row (the second difference - the ones that are all identical) by 2. (Because 2a=x, where x is the second difference, so x/2=a)
To work out b, use Algebra.
3a+b=y, where y is the difference between n=2 and n=1. You've already got a, so sub it in!
Finally, sub everything in and work out c by equating n=1 and a+b+c:
a+b+c=z, where z is the first term of the sequence.
If you don't understand or don't want to understand, just remember the steps and the highlighted bits. But if you still don't understand, email the Thinktivity Team and we will explain it (just tell us what you do not understant!).
And here are the same things you need to memorise for cubic sequences except for the fact that they come in this form:
an^3+bn^2+cn+d
And here are the four expressions you got to memorise and fill in:
1) a+b+c+d
2) 7a+3b+c
3) 12a+2b
4) 6a
The logic here is the same as above - if you want to know why, sub 1,2,3,4 into the general form and do some polynomial subtraction!
Don't understand? Don't hesitate to use the email box below and ask questions!
