While a linear pattern goes up by the exact same amount each step, a growing square or an accelerating rocket covers a larger difference with every move. A quadratic sequence is a sequence of numbers where the position-to-term rule can be written in the form . In this format, , , and are numbers, and cannot be zero.
Unlike an arithmetic pattern, a quadratic sequence does not have a constant first difference between consecutive terms. Instead, it is defined by a constant second difference, meaning the gaps between the gaps are always the exact same non-zero value.
Finding the formula for a quadratic sequence is a specific Higher Tier objective for OCR Mathematics. You must identify the component first, then determine the linear component that completes the sequence.
Breaking down a complex problem into familiar parts is often the most reliable way to solve it. OCR resources highly recommend using sequential reduction (often called the subtraction method) to find the full formula.
The relationship between the constant second difference and the term is fixed:
Once you divide the second difference by to find , you can calculate the sequence generated by . By subtracting these values from your original sequence, you are left with a linear remainder. This remainder will always form a standard linear sequence (an arithmetic sequence in the form ). You simply find the term of this linear remainder and add it to your term.
Calculate the formula for the term of the quadratic sequence: 4, 15, 30, 49, 72...
Step 1: Find the first and second differences to determine .
Step 2: Generate the sequence for .
Step 3: Subtract the sequence from the original sequence to find the linear remainder.
Step 4: Find the term of the linear remainder.
Step 5: Combine the quadratic and linear parts.
Students frequently use the full second difference as the coefficient . Remember that the second difference is equal to , so you must halve it.
Always set up a neat table for the subtraction phase (Original ) to avoid losing method marks due to simple negative number arithmetic slips.
Verify your final formula in the exam by substituting a higher value of (like ) to ensure it generates the correct original term.
OCR mark schemes specifically look for the linear remainder; showing the subtraction step clearly often secures an essential method mark (M1).
Quadratic sequence
A sequence of numbers where the term is a quadratic expression, characterised by having a constant second difference.
Position-to-term rule
An algebraic rule that allows you to calculate the value of any term in a sequence by substituting its position number ().
First difference
The numerical gap between two consecutive terms in a sequence.
Second difference
The difference between consecutive first differences in a sequence, which is always a non-zero constant for quadratic sequences.
Sequential reduction
A method for finding the term of a quadratic sequence by subtracting the quadratic part to reveal a simpler linear pattern.
Linear remainder
The arithmetic sequence produced by subtracting the term-values of from the original quadratic sequence.
Linear sequence
A sequence of numbers with a constant first difference, written in the form .
Put your knowledge into practice — try past paper questions for Mathematics
Quadratic sequence
A sequence of numbers where the term is a quadratic expression, characterised by having a constant second difference.
Position-to-term rule
An algebraic rule that allows you to calculate the value of any term in a sequence by substituting its position number ().
First difference
The numerical gap between two consecutive terms in a sequence.
Second difference
The difference between consecutive first differences in a sequence, which is always a non-zero constant for quadratic sequences.
Sequential reduction
A method for finding the term of a quadratic sequence by subtracting the quadratic part to reveal a simpler linear pattern.
Linear remainder
The arithmetic sequence produced by subtracting the term-values of from the original quadratic sequence.
Linear sequence
A sequence of numbers with a constant first difference, written in the form .