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Algebra · Sequences

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The nth term of a quadratic sequence

Why constant SECOND differences signal an n² rule, why the n²-coefficient is HALF the second difference, and the subtract-and-conquer routine that finds the full an² + bn + c.

Algebra · Sequences

The nth term of a quadratic sequence

Why constant SECOND differences signal an n² rule, why the n²-coefficient is HALF the second difference, and the subtract-and-conquer routine that finds the full an² + bn + c.

Why it works

When first differences aren't constant but keep changing by the same amount, the sequence is quadratic. For 3,8,15,24,353, 8, 15, 24, 35: differences 5,7,9,115, 7, 9, 11 — climbing by 2 each time. That constant second difference is the fingerprint of an n2n^2 term.

Why half? Watch n2n^2 itself: 1,4,9,16,251, 4, 9, 16, 25 has differences 3,5,7,93, 5, 7, 9 and second difference 2. Scaling to an2an^2 scales the second difference to 2a2a. So

a=second difference2a = \frac{\text{second difference}}{2}

— the second difference is DOUBLE the coefficient, and forgetting to halve (writing 2n22n^2 for a second difference of 2) is the standard error.

The subtract-and-conquer routine:
  1. Second difference → aa (halve it).
  2. Subtract an2an^2 from every term. What's left is a LINEAR sequence
(or a constant) — handle it with the linear toolkit.
  1. Add the pieces and check against two terms.
For 3,8,15,24,353, 8, 15, 24, 35: second difference 2, so a=1a = 1. Subtract n2n^2: 3−1,8−4,15−9,24−16,35−25=2,4,6,8,103-1, 8-4, 15-9, 24-16, 35-25 = 2, 4, 6, 8, 10 — that's 2n2n. So the nth term is n2+2nn^2 + 2n. Check n=4n = 4: 16+8=2416 + 8 = 24. ✓

For 5,11,21,35,535, 11, 21, 35, 53: differences 6,10,14,186, 10, 14, 18, second difference 4, so a=2a = 2. Subtract 2n22n^2: every term leaves exactly 33. Nth term: 2n2+32n^2 + 3.

The full three-piece case works the same. 4,11,22,37,564, 11, 22, 37, 56: second difference 4, a=2a = 2; subtracting 2n22n^2 leaves 2,3,4,5,6=n+12, 3, 4, 5, 6 = n + 1; so the nth term is 2n2+n+12n^2 + n + 1. Check n=5n = 5: 50+5+1=5650 + 5 + 1 = 56. ✓

Using and testing a quadratic rule. The 6th term of n2−3n+5n^2 - 3n + 5 is 36−18+5=2336 - 18 + 5 = 23 — brackets and order of operations as ever. And "is 150 a term of n2+5n^2 + 5?" becomes n2=145n^2 = 145; since 122=14412^2 = 144 and 132=16913^2 = 169, 145 is not a perfect square — no.