Quadratic and Surd Sequences

Understanding Quadratic Sequences

Imagine stacking cannonballs into a pyramid; the number of balls in each layer doesn't increase by a steady amount, but by an amount that itself is increasing. This is exactly how a quadratic sequence behaves.
A quadratic sequence is a sequence where the $n$ th term formula contains an $n^2$ term as the highest power. The formula is written in the general form $an^2 + bn + c$ , where $a$ , $b$ , and $c$ are constants and $a \neq 0$ .
Unlike linear sequences which have a constant first difference, quadratic sequences have a constant second difference. This is the difference between consecutive terms of the first difference sequence.
The coefficient of $n^2$ (the value of $a$ ) is always exactly half of this constant second difference.

Finding the $n$ th Term of a Quadratic Sequence

AQA frequently requires you to calculate the $n$ th term using a specific set of equations based on the sequence's differences. You must solve these three equations in order:

$2a = \text{Second Difference}$
$3a + b = \text{First Difference (between terms 1 and 2)}$
$a + b + c = \text{First Term } (U_1)$

Worked Example: Calculating a Quadratic $n$ th Term

Calculate the $n$ th term of the sequence $2, 9, 20, 35, 54 \dots$

Step 1: Find the first and second differences.

Sequence: $2, 9, 20, 35, 54$
First differences: $7, 11, 15, 19$
Second differences: $4, 4, 4$ (The constant second difference confirms it is a quadratic sequence).

Step 2: Use $2a$ to find $a$ .

$2a = 4 \implies a = 2$

Step 3: Use $3a + b$ to find $b$ .

Set this equal to the first difference between the 1st and 2nd term ( $7$ ).
$3(2) + b = 7 \implies 6 + b = 7 \implies b = 1$

Step 4: Use $a + b + c$ to find $c$ .

Set this equal to the first term in the original sequence ( $2$ ).
$2 + 1 + c = 2 \implies 3 + c = 2 \implies c = -1$

Step 5: Write the final formula.

Substitute $a$ , $b$ , and $c$ into $an^2 + bn + c$ .
$n$ th term = $2n^2 + n - 1$

Sequences Involving Surds

When a sequence involves square roots, the underlying pattern is often hidden until you strip away the complexity.
A surd is a root that results in an irrational number. Sequences containing surds can be arithmetic (adding a constant amount) or geometric (multiplying by a fixed common ratio, $r$ ).
To spot the pattern, radical simplification is the essential first step. You must factorize the number under the root into its largest square factor and a non-square factor, using the rule $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$ .
Once simplified, you can perform arithmetic with surds by adding or subtracting like surds (surds with the exact same number under the root).

Worked Example: Calculating the Next Term in a Surd Sequence

Identify the next term in the sequence $\sqrt{12}, \sqrt{27}, \sqrt{48}, \dots$

Step 1: Simplify the radicals.

$\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$
$\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}$
$\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}$

Step 2: Identify the pattern.

The simplified sequence is $2\sqrt{3}, 3\sqrt{3}, 4\sqrt{3}$ .
This is an arithmetic progression with a common difference of $\sqrt{3}$ .

Step 3: Find the next term and convert to exact form.

Add the common difference to the last term: $4\sqrt{3} + \sqrt{3} = 5\sqrt{3}$ .
AQA often requires answers converted back into a single radical exact form: $5\sqrt{3} = \sqrt{25 \times 3} = \sqrt{75}$ .

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Get full access to revision notes, key terms, and exam tips for every subtopic.

Exam Tips

Common Mistake
Students often forget to halve the second difference, incorrectly using the entire second difference ( $2a$ ) as the coefficient for the $n^2$ term instead of $a$ .
2
For questions with the command word 'Calculate', examiners award method marks (M1) for explicitly writing down the constant second difference and for the algebraic substitution to find $b$ and $c$ , so never skip these steps.
3
AQA requires answers for surd sequences to remain in exact form; converting your surds to decimals will result in lost marks.
4
To check if a specific number $X$ is a term in your quadratic sequence, set your formula equal to $X$ and solve for $n$ ; the sequence only contains the number if $n$ is a positive integer.

Key Terms(9)

Quadratic sequence: A sequence where the $n$ th term formula contains $n^2$ as the highest power and the second differences are constant.
First difference: The result of subtracting a term in a sequence from the term that immediately follows it.
Second difference: The difference between consecutive terms of the first difference sequence.
Coefficient of n²: The numerical multiplier of the $n^2$ term (denoted as $a$ ), which is always equal to half of the constant second difference.
Surd: A root (usually a square root) that results in an irrational number.
Common ratio: The fixed multiplier used to get from one term to the next in a geometric sequence.
Radical simplification: The process of rewriting a surd by factorizing the number under the root into its largest square factor and a non-square factor.
Arithmetic with surds: The process of adding, subtracting, multiplying, or dividing mathematical expressions containing surds.
Like surds: Surds that have the exact same radicand (the number under the root) and can therefore be directly added or subtracted.

Previous NoteArithmetic and Geometric Progressions Next Subtopic: Nth TermNth Term

Back to Special Sequences

Put your knowledge into practice — try past paper questions for Mathematics

Key Terms(9)

Quadratic sequence: A sequence where the $n$ th term formula contains $n^2$ as the highest power and the second differences are constant.
First difference: The result of subtracting a term in a sequence from the term that immediately follows it.
Second difference: The difference between consecutive terms of the first difference sequence.
Coefficient of n²: The numerical multiplier of the $n^2$ term (denoted as $a$ ), which is always equal to half of the constant second difference.
Surd: A root (usually a square root) that results in an irrational number.
Common ratio: The fixed multiplier used to get from one term to the next in a geometric sequence.
Radical simplification: The process of rewriting a surd by factorizing the number under the root into its largest square factor and a non-square factor.
Arithmetic with surds: The process of adding, subtracting, multiplying, or dividing mathematical expressions containing surds.
Like surds: Surds that have the exact same radicand (the number under the root) and can therefore be directly added or subtracted.

Quadratic and Surd Sequences

Understanding Quadratic Sequences

Imagine stacking cannonballs into a pyramid; the number of balls in each layer doesn't increase by a steady amount, but by an amount that itself is increasing. This is exactly how a quadratic sequence behaves.
A quadratic sequence is a sequence where the $n$ th term formula contains an $n^2$ term as the highest power. The formula is written in the general form $an^2 + bn + c$ , where $a$ , $b$ , and $c$ are constants and $a \neq 0$ .
Unlike linear sequences which have a constant first difference, quadratic sequences have a constant second difference. This is the difference between consecutive terms of the first difference sequence.
The coefficient of $n^2$ (the value of $a$ ) is always exactly half of this constant second difference.

Finding the $n$ th Term of a Quadratic Sequence

AQA frequently requires you to calculate the $n$ th term using a specific set of equations based on the sequence's differences. You must solve these three equations in order:

$2a = \text{Second Difference}$
$3a + b = \text{First Difference (between terms 1 and 2)}$
$a + b + c = \text{First Term } (U_1)$

Worked Example: Calculating a Quadratic $n$ th Term

Calculate the $n$ th term of the sequence $2, 9, 20, 35, 54 \dots$

Step 1: Find the first and second differences.

Sequence: $2, 9, 20, 35, 54$
First differences: $7, 11, 15, 19$
Second differences: $4, 4, 4$ (The constant second difference confirms it is a quadratic sequence).

Step 2: Use $2a$ to find $a$ .

$2a = 4 \implies a = 2$

Step 3: Use $3a + b$ to find $b$ .

Set this equal to the first difference between the 1st and 2nd term ( $7$ ).
$3(2) + b = 7 \implies 6 + b = 7 \implies b = 1$

Step 4: Use $a + b + c$ to find $c$ .

Set this equal to the first term in the original sequence ( $2$ ).
$2 + 1 + c = 2 \implies 3 + c = 2 \implies c = -1$

Step 5: Write the final formula.

Substitute $a$ , $b$ , and $c$ into $an^2 + bn + c$ .
$n$ th term = $2n^2 + n - 1$

Sequences Involving Surds

When a sequence involves square roots, the underlying pattern is often hidden until you strip away the complexity.
A surd is a root that results in an irrational number. Sequences containing surds can be arithmetic (adding a constant amount) or geometric (multiplying by a fixed common ratio, $r$ ).
To spot the pattern, radical simplification is the essential first step. You must factorize the number under the root into its largest square factor and a non-square factor, using the rule $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$ .
Once simplified, you can perform arithmetic with surds by adding or subtracting like surds (surds with the exact same number under the root).

Worked Example: Calculating the Next Term in a Surd Sequence

Identify the next term in the sequence $\sqrt{12}, \sqrt{27}, \sqrt{48}, \dots$

Step 1: Simplify the radicals.

$\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$
$\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}$
$\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}$

Step 2: Identify the pattern.

The simplified sequence is $2\sqrt{3}, 3\sqrt{3}, 4\sqrt{3}$ .
This is an arithmetic progression with a common difference of $\sqrt{3}$ .

Step 3: Find the next term and convert to exact form.

Add the common difference to the last term: $4\sqrt{3} + \sqrt{3} = 5\sqrt{3}$ .
AQA often requires answers converted back into a single radical exact form: $5\sqrt{3} = \sqrt{25 \times 3} = \sqrt{75}$ .

Sign up to continue reading

Get full access to revision notes, key terms, and exam tips for every subtopic.

Exam Tips

Common Mistake
Students often forget to halve the second difference, incorrectly using the entire second difference ( $2a$ ) as the coefficient for the $n^2$ term instead of $a$ .
2
For questions with the command word 'Calculate', examiners award method marks (M1) for explicitly writing down the constant second difference and for the algebraic substitution to find $b$ and $c$ , so never skip these steps.
3
AQA requires answers for surd sequences to remain in exact form; converting your surds to decimals will result in lost marks.
4
To check if a specific number $X$ is a term in your quadratic sequence, set your formula equal to $X$ and solve for $n$ ; the sequence only contains the number if $n$ is a positive integer.

Key Terms(9)

Quadratic sequence: A sequence where the $n$ th term formula contains $n^2$ as the highest power and the second differences are constant.
First difference: The result of subtracting a term in a sequence from the term that immediately follows it.
Second difference: The difference between consecutive terms of the first difference sequence.
Coefficient of n²: The numerical multiplier of the $n^2$ term (denoted as $a$ ), which is always equal to half of the constant second difference.
Surd: A root (usually a square root) that results in an irrational number.
Common ratio: The fixed multiplier used to get from one term to the next in a geometric sequence.
Radical simplification: The process of rewriting a surd by factorizing the number under the root into its largest square factor and a non-square factor.
Arithmetic with surds: The process of adding, subtracting, multiplying, or dividing mathematical expressions containing surds.
Like surds: Surds that have the exact same radicand (the number under the root) and can therefore be directly added or subtracted.

Previous NoteArithmetic and Geometric Progressions Next Subtopic: Nth TermNth Term

Back to Special Sequences

Put your knowledge into practice — try past paper questions for Mathematics

Key Terms(9)

Quadratic sequence: A sequence where the $n$ th term formula contains $n^2$ as the highest power and the second differences are constant.
First difference: The result of subtracting a term in a sequence from the term that immediately follows it.
Second difference: The difference between consecutive terms of the first difference sequence.
Coefficient of n²: The numerical multiplier of the $n^2$ term (denoted as $a$ ), which is always equal to half of the constant second difference.
Surd: A root (usually a square root) that results in an irrational number.
Common ratio: The fixed multiplier used to get from one term to the next in a geometric sequence.
Radical simplification: The process of rewriting a surd by factorizing the number under the root into its largest square factor and a non-square factor.
Arithmetic with surds: The process of adding, subtracting, multiplying, or dividing mathematical expressions containing surds.
Like surds: Surds that have the exact same radicand (the number under the root) and can therefore be directly added or subtracted.

Quadratic and Surd Sequences

Understanding Quadratic Sequences

Finding the nnnth Term of a Quadratic Sequence

Worked Example: Calculating a Quadratic nnnth Term

Sequences Involving Surds

Worked Example: Calculating the Next Term in a Surd Sequence

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Exam Tips

Key Terms(9)

Key Terms(9)

Quadratic and Surd Sequences

Understanding Quadratic Sequences

Finding the nnnth Term of a Quadratic Sequence

Worked Example: Calculating a Quadratic nnnth Term

Sequences Involving Surds

Worked Example: Calculating the Next Term in a Surd Sequence

Sign up to continue reading

Exam Tips

Key Terms(9)

Key Terms(9)

Finding the $n$ th Term of a Quadratic Sequence

Worked Example: Calculating a Quadratic $n$ th Term

Finding the $n$ th Term of a Quadratic Sequence

Worked Example: Calculating a Quadratic $n$ th Term