Quadratic Sequences

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