What is a Quadratic Sequence?
In science and mathematics, we often look for patterns in data. Some patterns are simple and grow by the same amount each time , these are linear sequences. But many real-world patterns grow faster and faster, and these often follow a quadratic sequence.
Quadratic Sequence: A number sequence in which the second differences are constant (always the same value). The general term is given by a formula containing , making it a second-degree (quadratic) expression.
Think about how the area of a square grows as its side length increases: a square has area 1, a has area 4, a has area 9. The areas form the sequence , a classic quadratic sequence!
Imagine rolling a ball down a hill. Each second, it doesn't just move faster , it accelerates. The distance it covers each second keeps increasing. If you recorded the total distance at each second, you'd get a quadratic sequence!
First and Second Differences
The key tool for identifying and working with quadratic sequences is the idea of differences.
First differences are found by subtracting each term from the next:
Second differences are found by subtracting each first difference from the next:
For a quadratic sequence, the second differences are always equal (constant).
Example: Analyse the sequence
Step 1 , Write out the terms:
Step 2 , Find first differences:
Step 3 , Find second differences:
The second differences are constant (+2), so this is a quadratic sequence. ✓
If the first differences are constant, the sequence is linear, not quadratic. Always check second differences before concluding a sequence is quadratic.
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