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Lesson overview

Algebra — sequences and patterns

A quadratic sequence has a constant second difference. The first differences are the gaps between consecutive terms. The second differences are the gaps between the first differences. If the second difference is constant, the nth term formula is quadratic ( $an^2 + bn + c$ ). To find the nth term: (1) Calculate first differences, (2) Calculate second differences, (3) If constant, the leading coefficient $a = \frac{\text{second difference}}{2}$ , (4) Use the pattern to find $b$ and $c$ .

Identifying a quadratic sequence using second differences

Is the sequence 3, 8, 15, 24, 35 quadratic?

Term	Value
1	3
2	8
3	15
4	24
5	35

VerdictYes — second difference is constant (2), so it's quadratic

Constant second differences = quadratic sequence. If second differences were different, it would be higher order.

Finding the nth term formula step by step

Find the nth term formula for 2, 6, 12, 20, 30.

$n$	1	2	3	4	5
Term	2	6	12	20	30

The leading coefficient is half the second difference: $a = 2 ÷ 2 = 1$ . Then test values to find the rest.

Verifying the formula with known terms

Check that $n^2 + n$ generates the sequence 2, 6, 12, 20, 30.

ConclusionFormula verified! We can now find any term.

Always substitute back into your formula to check it works for the original sequence.

Worked example — find the nth term and use it

The sequence 1, 5, 11, 19, 29 is quadratic. Find the nth term formula, then find the 10th term.

$n$	1	2	3	4	5
Term	1	5	11	19	29
1st diff	–	4	6	8	10
2nd diff	–	–	2	2	2

Write out the sequence and calculate first differences: 4, 6, 8, 10.
Calculate second differences: 2, 2, 2 (constant, so it's quadratic).
Find the leading coefficient: $a = \frac{2}{2} = 1$ .
Substitute two terms into $n^2 + bn + c$ to find $b$ and $c$ .
From $n=1$ : $1 + b + c = 1$ , so $b + c = 0$ .
From $n=2$ : $4 + 2b + c = 5$ , so $2b + c = 1$ .
Subtract to solve: $b = 1$ and $c = -1$ .
The formula is $a_n = n^2 + n - 1$ . For $n=10$ : $a_{10} = 109$ .

Try it

Find the nth term formula for each sequence using second differences, then verify it.

Question 1

The sequence 1, 4, 9, 16, 25 has first differences 3, 5, 7, 9. What is the second difference?

💡 Second difference = difference of first differences: 5 − 3 = 2, 7 − 5 = 2, etc.

Question 2

A sequence has a constant second difference of 6. What is the leading coefficient $a$ in $an^2 + bn + c$ ?

💡 $a = \frac{\text{second difference}}{2} = \frac{6}{2} = 3$ .

Question 3

The nth term of a sequence is $2n^2 + 3$ . What is the 4th term?

💡 $a_4 = 2(4)^2 + 3 = 2 \times 16 + 3 = 32 + 3 = 35$ .

Question 4

Which sequence is quadratic?

💡 The sequence of perfect squares has constant second difference (2). The others are linear or exponential.

Question 5

The sequence 2, 7, 14, 23, 34 has nth term $n^2 + 2n - 1$ . Find the 6th term.

💡 $a_6 = 6^2 + 2(6) - 1 = 36 + 12 - 1 = 47$ .

Common mistakes

Watch for these when working through the lesson.

Confusing first differences with second differences: second differences must be constant for a quadratic sequence.
Forgetting to divide the second difference by 2 to find $a$ : the formula is $a = \frac{\text{2nd difference}}{2}$ .
Not checking the formula against known terms: always substitute at least two values to verify $b$ and $c$ .

Derive the nth term of a quadratic sequence from first differences

Lesson overview

Worked example — find the nth term and use it

Try it

Common mistakes

Related topics

Independent practice will plug in here