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Quadratic Sequence — Definition, Formula & Examples

A quadratic sequence is a sequence of numbers where the second differences between consecutive terms are constant. The nth term is defined by a quadratic expression of the form an2+bn+can^2 + bn + c.

A sequence {Tn}\{T_n\} is quadratic if its general term can be written as Tn=an2+bn+cT_n = an^2 + bn + c where a≠0a \neq 0 and aa, bb, cc are constants. Equivalently, the first differences Tn+1−TnT_{n+1} - T_n form an arithmetic (linear) sequence, so the second differences Δ2Tn\Delta^2 T_n are constant and equal to 2a2a.

Key Formula

Tn=an2+bn+cT_n = an^2 + bn + c
Where:
  • TnT_n = The nth term of the sequence
  • aa = Half the constant second difference (a ≠ 0)
  • bb = Linear coefficient
  • cc = Constant term

How It Works

To identify a quadratic sequence, compute the first differences (subtract each term from the next). Then compute the second differences (subtract each first difference from the next). If the second differences are all the same constant, the sequence is quadratic. To find the nth-term rule, use the fact that the second difference equals 2a2a to determine aa, then solve for bb and cc using known terms.

Worked Example

Problem: Find the nth-term formula for the sequence 3, 10, 21, 36, 55, …
Find first differences: Subtract consecutive terms.
10−3=7,21−10=11,36−21=15,55−36=1910-3=7,\quad 21-10=11,\quad 36-21=15,\quad 55-36=19
Find second differences: Subtract consecutive first differences. They are all 4, confirming a quadratic sequence.
11−7=4,15−11=4,19−15=411-7=4,\quad 15-11=4,\quad 19-15=4
Determine a: The second difference equals 2a.
2a=4  ⟹  a=22a = 4 \implies a = 2
Find b and c: Use T₁ = 3 and T₂ = 10. Substituting n = 1: 2(1) + b(1) + c = 3, so b + c = 1. Substituting n = 2: 2(4) + 2b + c = 10, so 2b + c = 2.
b+c=1and2b+c=2  ⟹  b=1,  c=0b + c = 1 \quad\text{and}\quad 2b + c = 2 \implies b = 1,\; c = 0
Answer: Tn=2n2+nT_n = 2n^2 + n

Why It Matters

Quadratic sequences appear frequently on GCSE and A-level exams and in SAT/ACT pattern-recognition problems. Understanding them also builds the foundation for polynomial regression and modeling real-world data such as projectile heights or areas that grow non-linearly.

Common Mistakes

Mistake: Using first differences instead of second differences to test for a quadratic pattern.
Correction: Constant first differences indicate a linear (arithmetic) sequence. For a quadratic sequence, the first differences change but the second differences are constant.

Related Terms