Finding the nth Term of a Quadratic Sequence
⭐ Higher Tier Content
A quadratic sequence is a sequence where the differences between terms are not constant, but the second differences are constant. Finding the nth term means writing a formula that allows any term in the sequence to be calculated by substituting a value for \( n \).
The nth term of a quadratic sequence has the form:
$$
an^2 + bn + c
$$
where \( a \), \( b \) and \( c \) are constants.
Recognising a Quadratic Sequence
Consider the sequence:
$$
1,\ 4,\ 9,\ 16
$$
Find the first differences:
$$
3,\ 5,\ 7
$$
Find the second differences:
$$
2,\ 2
$$
Because the second differences are constant, the sequence is quadratic.
Finding the Value of a
The value of \( a \) is found using the second difference.
For quadratic sequences:
$$
a = \frac{\text{second difference}}{2}
$$
In the example above, the second difference is \( 2 \), so:
$$
a = 1
$$
Start with:
$$
n^2 + bn + c
$$
Finding the Values of b and c
Use known terms from the sequence to form equations.
Using the sequence:
$$
1,\ 4,\ 9,\ 16
$$
When \( n = 1 \), the term is \( 1 \):
$$
1 + b + c = 1
$$
$$
b + c = 0
$$
When \( n = 2 \), the term is \( 4 \):
$$
4 + 2b + c = 4
$$
$$
2b + c = 0
$$
Subtract the first equation from the second:
$$
b = 0
$$
So:
$$
c = 0
$$
The nth term is:
$$
n^2
$$
Another Numerical Example
Consider the sequence:
$$
2,\ 7,\ 16,\ 29
$$
First differences:
$$
5,\ 9,\ 13
$$
Second differences:
$$
4,\ 4
$$
So:
$$
a = 2
$$
Start with:
$$
2n^2 + bn + c
$$
When \( n = 1 \), term \( = 2 \):
$$
2 + b + c = 2
$$
$$
b + c = 0
$$
When \( n = 2 \), term \( = 7 \):
$$
8 + 2b + c = 7
$$
$$
2b + c = -1
$$
Subtract:
$$
b = -1
$$
So:
$$
c = 1
$$
The nth term is:
$$
2n^2 - n + 1
$$
Finding the nth Term from a Diagram
Diagrammatic sequences often show shapes increasing in a quadratic pattern, such as squares made of dots.
If the number of items forms a quadratic sequence:
• write down the number of items in each diagram
• find first and second differences
• follow the same method to find \( a \), \( b \) and \( c \)
The process is identical once the numerical sequence is known.
Checking the Rule
Always check the nth term by substituting values of \( n \).
For example, using:
$$
2n^2 - n + 1
$$
When \( n = 3 \):
$$
2(9) - 3 + 1 = 16
$$
This matches the third term, confirming the rule is correct.
Key Points to Remember
Quadratic sequences have constant second differences.
The nth term has the form \( an^2 + bn + c \).
The value of \( a \) is half the second difference.
Use known terms to find \( b \) and \( c \).
Always check the rule against the sequence.
Finding the nth term of a quadratic sequence is a key skill that links patterns, algebra and graphs.