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Understanding Quadratic Sequences

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 $an^2 + bn + c$ . In this format, $a$ , $b$ , and are numbers, and cannot be zero.

The Subtraction Method

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 $an^2 + bn + c$ formula.

The relationship between the constant second difference and the $n^2$ term is fixed:

2a = \text{second difference}

Once you divide the second difference by $2$ to find $a$ , you can calculate the sequence generated by $an^2$ . By subtracting these $an^2$ 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.

Worked Example: Calculating the Formula

Calculate the formula for the $n^{\text{th}}$ term of the quadratic sequence: 4, 15, 30, 49, 72...

Step 1: Find the first and second differences to determine $a$ .

Sequence: $4, 15, 30, 49, 72$
First differences: $+11, +15, +19, +23$
Second differences: $+4, +4, +4$

Step 2: Generate the $2n^2$ sequence for $n = 1, 2, 3, 4, 5$ .

$2(1)^2 = 2$
$2(2)^2 = 8$
$2 ($

Step 3: Subtract the $2n^2$ sequence from the original sequence to find the linear remainder.

Original sequence: $4, 15, 30, 49, 72$
Subtract $2n^2$ : $2, 8, 18, 32, 50$

Step 4: Find the $n^{\text{th}}$ term of the linear remainder.

The remainder sequence goes up by $+5$ each time, so it starts with $5n$ .
The "zero term" (the term before $n=1$ ) is $2 - 5 = -3$ .
The linear remainder formula is .

Step 5: Combine the quadratic and linear parts.

Final formula: $2n^2 + 5n - 3$

Students frequently use the full second difference as the coefficient $a$ . Remember that the second difference is equal to $2a$ , so you must halve it.

Always set up a neat table for the subtraction phase (Original $- an^2$ ) 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 $n$ (like $n=4$ ) 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).

A sequence of numbers where the $n^{\text{th}}$ term is a quadratic expression, characterised by having a constant second difference.

An algebraic rule that allows you to calculate the value of any term in a sequence by substituting its position number ( $n$ ).

The numerical gap between two consecutive terms in a sequence.

The difference between consecutive first differences in a sequence, which is always a non-zero constant for quadratic sequences.

A method for finding the $n^{\text{th}}$ term of a quadratic sequence by subtracting the quadratic part to reveal a simpler linear pattern.

The arithmetic sequence produced by subtracting the term-values of $an^2$ from the original quadratic sequence.

A sequence of numbers with a constant first difference, written in the form $bn + c$ .