Finding the nth term of quadratic sequences might seem tricky...
Maths391 views·Updated 21 Jul 2026·2 pagesUnderstanding the nth Term in Math
Saarah@saarahs.3
1
of 2
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.
2
of 2
Finding the nth Term Formula
Here's the brilliant shortcut: half the second difference and that becomes your coefficient of n². In the sequence 3, 12, 27, 48, 75, 108, the second differences are all 6.
Half of 6 is 3, so the nth term is 3n². It's that straightforward! If your second difference was 10, you'd get 5n² as your nth term.
Sometimes you need to add a bit more to get the complete formula. Look at the sequence 2, 5, 10, 17, 26 - the second differences are 2, giving us n² as the base.
Remember: Start with n², then adjust by comparing your formula to the actual sequence values.
But when you check n² against the original sequence (1, 4, 9, 16, 25), you notice the original starts with 2, not 1. So you need n² + 1 to get the perfect match!
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Maths391 views·Updated 21 Jul 2026·2 pagesUnderstanding the nth Term in Math
Saarah@saarahs.3
Finding the nth term of quadratic sequences might seem tricky at first, but there's actually a simple pattern you can follow. Once you master the difference method, you'll be able to tackle any quadratic sequence with confidence.
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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.
2
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Finding the nth Term Formula
Here's the brilliant shortcut: half the second difference and that becomes your coefficient of n². In the sequence 3, 12, 27, 48, 75, 108, the second differences are all 6.
Half of 6 is 3, so the nth term is 3n². It's that straightforward! If your second difference was 10, you'd get 5n² as your nth term.
Sometimes you need to add a bit more to get the complete formula. Look at the sequence 2, 5, 10, 17, 26 - the second differences are 2, giving us n² as the base.
Remember: Start with n², then adjust by comparing your formula to the actual sequence values.
But when you check n² against the original sequence (1, 4, 9, 16, 25), you notice the original starts with 2, not 1. So you need n² + 1 to get the perfect match!
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