Step by step
The question
Question: 7, 9, 9, 7, 3, …: find the rule for term n.
1 First differences
We write the terms in a row.
Under each gap, the jump from one term to the next: 2, 0, −2, −4.
The jumps are not all the same.
So this is not an arithmetic sequence.
2 Second differences
Now the jumps of the jumps: −2 every time.
They are the same.
The same second difference means the rule has n squared in it.
A list like this is a quadratic sequence.
3 Find a
The number in front of n squared is a.
a is half the second difference: −2 divided by 2 is −1.
Why half?
Look at n squared itself: 1, 4, 9, 16.
The jumps are 3, 5, 7, and their jumps are 2.
So n squared gives a second difference of 2.
Careful!
a is not the second difference itself.
It is half of it.
4 Take away −n²
A table: n is the term number.
Under each term: −1 times n squared.
−1, −4, −9 and so on.
Take it away from each term.
What is left: 8, 13, 18, 23, 28.
5 The linear part
What is left goes up by 5 each time: an arithmetic sequence.
Its first term is 8.
The arithmetic rule: the first term, plus (n − 1) times the difference.
Open the bracket and put the plain numbers together: 5n + 3.
6 Put it together
The rule is −1 n squared, plus what was left: −n² + 5n + 3.
7 Check
Let's check the rule with n = 1, 2 and 3.
We get 7, 9, 9: the first terms of the list.
So the rule is right!
Let's look back.
First: first differences.
Then: second differences.
Then: find a.
Then: take away −n².
Then: the linear part.
Then: put it together.
Last: check.