Worked examples
The question
Question: 2, 6, 12, 20, 30, …: find the rule for term n.
2, 6, 12, 20, 30, … find the nth term
1 First differences
We write the terms in a row.
Under each gap, the jump from one term to the next: 4, 6, 8, 10.
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.
= same 2nd difference → the rule has n²
3 Find a
The number in front of n squared is a.
a is half the second difference: 2 divided by 2 is 1.
a = 22 = 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.
n²: 1, 4, 9, 16 → +3, +5, +7 → 2
Careful!
a is not the second difference itself.
It is half of it.
Careful: not a = 2 ✗
4 Take away n²
A table: n is the term number.
Under each term: n squared.
1, 4, 9 and so on.
Take it away from each term.
What is left: 1, 2, 3, 4, 5.
5 The linear part
What is left goes up by 1 each time: an arithmetic sequence.
Its first term is 1.
The arithmetic rule: the first term, plus (n − 1) times the difference.
what is left = 1 + (n − 1) × 1
aₙ = a₁ + (n − 1) × d
Open the bracket and put the plain numbers together: n.
= n
6 Put it together
The rule is n squared, plus what was left: n² + n.
an = n2 + n
7 Check
Let's check the rule with n = 1, 2 and 3.
We get 2, 6, 12: the first terms of the list.
So the rule is right!
n = 1: 2, n = 2: 6, n = 3: 12
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.
▶ Watch it solved
The question
Question: 4, 10, 18, 28, 40, …: find the rule for term n.
4, 10, 18, 28, 40, … find the nth term
1 First differences
We write the terms in a row.
Under each gap, the jump from one term to the next: 6, 8, 10, 12.
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.
= same 2nd difference → the rule has n²
3 Find a
The number in front of n squared is a.
a is half the second difference: 2 divided by 2 is 1.
a = 22 = 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.
n²: 1, 4, 9, 16 → +3, +5, +7 → 2
Careful!
a is not the second difference itself.
It is half of it.
Careful: not a = 2 ✗
4 Take away n²
A table: n is the term number.
Under each term: n squared.
1, 4, 9 and so on.
Take it away from each term.
What is left: 3, 6, 9, 12, 15.
5 The linear part
What is left goes up by 3 each time: an arithmetic sequence.
Its first term is 3.
The arithmetic rule: the first term, plus (n − 1) times the difference.
what is left = 3 + (n − 1) × 3
aₙ = a₁ + (n − 1) × d
Open the bracket and put the plain numbers together: 3n.
= 3n
6 Put it together
The rule is n squared, plus what was left: n² + 3n.
an = n2 + 3n
7 Check
Let's check the rule with n = 1, 2 and 3.
We get 4, 10, 18: the first terms of the list.
So the rule is right!
n = 1: 4, n = 2: 10, n = 3: 18
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.
▶ Watch it solved
The question
Question: 0, 6, 16, 30, 48, …: find the rule for term n.
0, 6, 16, 30, 48, … find the nth term
1 First differences
We write the terms in a row.
Under each gap, the jump from one term to the next: 6, 10, 14, 18.
The jumps are not all the same.
So this is not an arithmetic sequence.
2 Second differences
Now the jumps of the jumps: 4 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.
= same 2nd difference → the rule has n²
3 Find a
The number in front of n squared is a.
a is half the second difference: 4 divided by 2 is 2.
a = 42 = 2
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.
n²: 1, 4, 9, 16 → +3, +5, +7 → 2
Careful!
a is not the second difference itself.
It is half of it.
Careful: not a = 4 ✗
4 Take away 2n²
A table: n is the term number.
Under each term: 2 times n squared.
2, 8, 18 and so on.
Take it away from each term.
What is left: −2, −2, −2, −2, −2.
5 The linear part
What is left is −2 every time.
So that part is just −2.
what is left = −2
6 Put it together
The rule is 2 n squared, plus what was left: 2n² − 2.
an = 2n2 − 2
7 Check
Let's check the rule with n = 1, 2 and 3.
We get 0, 6, 16: the first terms of the list.
So the rule is right!
n = 1: 0, n = 2: 6, n = 3: 16
Let's look back.
First: first differences.
Then: second differences.
Then: find a.
Then: take away 2n².
Then: the linear part.
Then: put it together.
Last: check.
▶ Watch it solved
