Concept 14 of 19Advanced
Watch on YouTubeVideoSecond-differences: spotting quadratic patterns
When first differences AREN'T constant, look at second differences (differences of differences).
If second differences are constant, the sequence is quadratic: aₙ = An² + Bn + C.
Example: 2, 6, 12, 20, 30
First differences: 4, 6, 8, 10 (arithmetic)
Second differences: 2, 2, 2 (constant) → quadratic
To find A, B, C, use three known terms as equations.
Example
Triangular numbers 1, 3, 6, 10, 15 have first diffs 2,3,4,5 and second diffs 1,1,1,… Confirms Tₙ is quadratic: Tₙ = n(n+1)/2 = ½n² + ½n.
💡 Tip:Three levels of constant differences → cubic. Four → quartic. And so on.
▸Why does this work? (derivation)
Why? For aₙ = An²+Bn+C: aₙ₊₁ − aₙ = 2An + (A+B), which is itself linear in n. The differences of THIS are 2A — a constant! So constant second differences signal quadratic.
🎯 Try it!
5 questions to check what you just read.
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- Q1.First differences of 3, 5, 8, 12, 17:
- Q2.Second differences of 1, 4, 9, 16, 25:
- Q3.Type of sequence 1, 4, 9, 16, 25:
- Q4.Second diffs of 2, 8, 16, 26, 38:
- Q5.If the 2nd differences of a sequence are constant, the sequence is: