Step by step
The question
Question: 4, 13, 28, 49, 76, …: find term number 12.
1 First differences
We write the terms in a row.
Under each gap, the jump from one term to the next: 9, 15, 21, 27.
The jumps are not all the same.
So this is not an arithmetic sequence.
2 Second differences
Now the jumps of the jumps: 6 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: 6 divided by 2 is 3.
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 3n²
A table: n is the term number.
Under each term: 3 times n squared.
3, 12, 27 and so on.
Take it away from each term.
What is left: 1, 1, 1, 1, 1.
5 The linear part
What is left is 1 every time.
So that part is just 1.
6 Put it together
The rule is 3 n squared, plus what was left: 3n² + 1.
7 Find the 12th term
Put n = 12 in the rule.
That is 433.
8 Check
Let's check the rule with n = 1, 2 and 3.
We get 4, 13, 28: 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 3n².
Then: the linear part.
Then: put it together.
Then: find the 12th term.
Last: check.