Understanding Quadratic Sequences Through Differences
Quadratic sequences follow a special pattern that becomes clear when you look at their differences. Take the sequence 1, 4, 9, 16, 25, 36 - these are perfect squares, but let's see how the difference method works.
When you find the first difference (subtracting each term from the next), you get: 3, 5, 7, 9, 11. Notice these aren't the same, so we need to go deeper.
The second difference gives us: 2, 2, 2, 2. When the second differences are constant, you've got a quadratic sequence! This particular sequence has the nth term of n².
Quick Tip: If your second differences are all the same number, you're definitely dealing with a quadratic sequence.
The pattern becomes clearer with another example: 2, 8, 18, 32, 50, 72. The second differences here are all 4, and the nth term turns out to be 2n².
