Quadratic Sequences Revision Notes for Leaving Cert Mathematics

Quadratic Sequences

What is a quadratic sequence?

Quadratic sequences are sequences where the nth term contains n² as the highest power. These sequences follow a specific pattern that can be identified using a mathematical technique called finding differences.

The general form of a quadratic sequence is: $T_n = an^2 + bn + c$

where a, b, and c are constants, and a ≠ 0.

Key characteristic: constant second differences

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The most important feature that identifies a quadratic sequence is that the second differences are always constant. This means when you find the differences between consecutive terms, then find the differences of those differences, you get the same number throughout.

In the example above, the sequence 1, 6, 15, 28, 45 has:

First differences: 5, 9, 13, 17
Second differences: 4, 4, 4 (constant)

Since the second differences are constant, this confirms the sequence is quadratic.

The key rule for quadratic sequences

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Remember: In a quadratic sequence, the coefficient of n² in the nth term is half the second difference.

If the second difference is 4, then the coefficient of n² is 2
If the second difference is 6, then the coefficient of n² is 3

Finding the nth term formula

To find the formula for the nth term of a quadratic sequence, follow these steps:

Step 1: Create a differences table

Step 2: Use the general form

The nth term will be: $T_n = an^2 + bn + c$

Where a = half the second difference

Step 3: Find the values of a, b, and c

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Worked Example: Finding the nth term

Find the nth term for the sequence: 3, 10, 21, 36

Solution:

Step 1: Find the differences:

First differences: 7, 11, 15
Second differences: 4, 4

Step 2: Since the second difference is 4: a = 4 ÷ 2 = 2

Step 3: So far we have: $T_n = 2n^2 + bn + c$

Step 4: Use the first two terms to find b and c:

When n = 1: $T_1 = 2(1)^2 + b(1) + c = 2 + b + c = 3$
So: b + c = 1 ... (equation 1)
When n = 2: $T_2 = 2(2)^2 + b(2) + c = 8 + 2b + c = 10$
So: 2b + c = 2 ... (equation 2)

Step 5: Solve the simultaneous equations:

From equation 1: b + c = 1
From equation 2: 2b + c = 2
Subtract: b = 1
Substitute back: 1 + c = 1, so c = 0

Step 6: Therefore: $T_n = 2n^2 + n$

Pattern recognition in visual sequences

Quadratic sequences often appear in practical contexts, such as tiling patterns or geometric arrangements.

In the tiling example, we can see that:

Pattern 1 has 21 blue tiles
Pattern 2 has 33 blue tiles

To find the pattern, we would:

Complete the table for patterns 3, 4, and 5
Find the differences between consecutive terms
Check if the second differences are constant
Use the method above to find the nth term formula

Common exam tips

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Essential Exam Strategies:

Always check second differences first - if they're constant, you have a quadratic sequence
Remember the key relationship: coefficient of n² = ½ × second difference
Show your working clearly in difference tables
Substitute back to check your formula works for the original terms
Watch out for sequences that might look quadratic but aren't - always verify with second differences

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Worked Example: Complete Solution

Find the nth term for: 5, 8, 13, 20, 29

Solution:

Step 1: Find the differences:

First differences: 3, 5, 7, 9
Second differences: 2, 2, 2

Step 2: Since second difference = 2, then a = 1

Step 3: Formula form: $T_n = n^2 + bn + c$

Step 4: Using first two terms:

$T_1 = 1 + b + c = 5$ , so b + c = 4
$T_2 = 4 + 2b + c = 8$ , so 2b + c = 4

Step 5: Solving: b = 0, c = 4

Step 6: Therefore: $T_n = n^2 + 4$

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Key Points to Remember:

Quadratic sequences have n² as the highest power in their nth term formula
Second differences are constant in quadratic sequences - this is the key identifier
The coefficient of n² equals half the second difference - use this rule to start finding the formula
Always use a differences table to organise your working systematically
Check your answer by substituting values back into your formula

Quadratic Sequences (Leaving Cert Mathematics): Revision Notes