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9 Jan 2026

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2 pages

Understanding the nth Term in Math

Saarah

@saarahs.3

Finding the nth term of quadratic sequences might seem tricky... Show more

1 4 9 16 25 36 1st difference

3 5 7 9 "

2222

nd
2
difference

nth term for this sequence: 72

Example 2

28 18 32 50 72

1st difference

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.

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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9 Jan 2026

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2 pages

Understanding 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

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.

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By signing up you accept Terms of Service and Privacy Policy

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!