Finding the nth term of quadratic sequences might seem tricky...
Maths393 views·Updated 11 Aug 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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Maths393 views·Updated 11 Aug 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
of 2Sign up to see the content. It's free!
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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!
We thought you’d never ask...
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
You can download the app from Google Play Store and Apple App Store.
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