Sequences — AQA GCSE Mathematics

Sequences are ordered lists of numbers that follow a specific pattern or rule. In AQA GCSE Mathematics, you will learn to identify patterns, continue sequences, and find general expressions for the nth term of linear and quadratic sequences. You will also explore special sequences such as Fibonacci, square numbers, cube numbers, and triangular numbers.

This topic is fundamental because it develops algebraic reasoning and pattern recognition, which are essential for higher-tier topics like calculus and series. Sequences appear in many real-world contexts, from savings plans to population growth, and exam questions often test your ability to apply rules, solve equations, and justify whether a number belongs to a sequence.

• →A sequence is an ordered list of terms, where each term has a position number n. The nth term rule allows you to calculate any term without listing all previous terms.
• →For a linear sequence, the difference between consecutive terms is constant. The nth term is of the form an + b, where a is the common difference and b is found by substituting n = 1.
• →For a quadratic sequence, the second difference is constant. The nth term is of the form an squared + bn + c, where a is half the second difference, and b and c are found using simultaneous equations or by subtracting an squared.
• →Special sequences include Fibonacci (each term is the sum of the two previous terms), square numbers (n squared), cube numbers (n cubed), and triangular numbers (n(n+1)/2).
• →You must be able to determine whether a given number is a term of a sequence by setting the nth term equal to that number and solving for n; n must be a positive integer.

• Correctly generate terms by repeatedly applying a given term-to-term rule, such as adding or multiplying by a constant.
• Correctly generate terms by substituting n = 1, 2, 3, ... into a given position-to-term rule.
• Distinguish between a term-to-term rule and a position-to-term rule and use the appropriate method.
• Generate a specified term, such as the 10th term, directly from a position-to-term rule without listing all previous terms.
• Recognise and continue the sequence of triangular numbers: 1, 3, 6, 10, 15, ...
• Recognise and continue the sequence of square numbers: 1, 4, 9, 16, 25, ...
• Recognise and continue the sequence of cube numbers: 1, 8, 27, 64, 125, ...
• Identify a simple arithmetic progression and use its common difference to find further terms or the nth term.
• Recognises a Fibonacci-type sequence as one built by adding the two preceding terms, and continues it correctly from any given starting pair.
• Recognises a quadratic sequence from a constant second difference and continues the sequence to find the next terms.
• Identifies a simple geometric progression by a constant multiplier r, where r is a rational number greater than 0, and continues it by repeated multiplication.
• Uses the form rⁿ with integer n to generate or check terms of a geometric progression, including fractional ratios such as r = 1/2.
• Distinguishes the three families when several are offered, for example rejecting an arithmetic sequence as Fibonacci-type or quadratic.
• Applies the correct rule to find a named term, such as the 10th term, rather than only the next term.
• Recognises that a geometric progression may have a surd common ratio $r$, such as $\sqrt{2}$ or $\sqrt{3}$, and that $r$ is then irrational rather than rational.
• Continues a surd-ratio sequence exactly, for example $1, \sqrt{2}, 2, 2\sqrt{2}, 4$ with $r = \sqrt{2}$.
• Simplifies products of surds correctly, including $\sqrt{2} \times \sqrt{2} = 2$ and $\sqrt{3} \times \sqrt{3} = 3$.
• Uses $ar^{n-1}$ to find a stated term, for example the 4th term of a sequence with first term $a$ and $r = \sqrt{2}$ is $2a\sqrt{2}$.
• Keeps answers in exact surd form, or converts to a decimal only when the question requests an approximation.
• Distinguishes a surd-ratio geometric progression from a quadratic sequence by checking for a constant multiplier rather than a constant second difference.
• Find the common difference between consecutive terms and use it as the coefficient of n.
• Compare the first term with the value given by the coefficient of n when n = 1, then adjust by adding or subtracting a constant.
• Form the expression in the form an + b, where a is the common difference and b is the constant adjustment.
• Verify the expression by substituting n = 1, 2 and 3 and checking the results match the given sequence.
• Calculate the first differences between consecutive terms and then the constant second difference.
• Divide the constant second difference by 2 to determine a, the coefficient of n².
• Subtract the an² component from each original term to generate a linear remainder sequence.
• Determine the nth term of the linear remainder sequence (bn + c).
• Combine the quadratic and linear parts to state the full nth term and verify by substitution.

• 💡Read the question carefully to identify whether you are given a term-to-term rule or a position-to-term rule, and use the matching method.
• 💡For a position-to-term rule, write down the substitution for each required term, especially when the expression involves negative numbers or powers.
• 💡Check your generated terms by testing whether they fit the rule; for a term-to-term rule, verify that the difference or ratio is consistent.
• 💡Write out the first few terms of the sequence to help you recognise the pattern before answering.
• 💡For arithmetic progressions, find the common difference by subtracting consecutive terms, and use it to extend the sequence or find a formula.
• 💡Check whether a sequence is triangular, square or cube by comparing its terms with the standard lists you have learned.
• 💡Write the first differences, and then the second differences, before deciding whether a sequence is arithmetic, quadratic or neither.
• 💡For a geometric progression, divide a term by the previous term to confirm the multiplier r is constant before using rⁿ.
• 💡Check a Fibonacci-type continuation by confirming each new term equals the sum of the two terms immediately before it.
• 💡Divide each term by the previous term to confirm the surd multiplier is constant before continuing the sequence.
• 💡Simplify surd products step by step, using $\sqrt{a} \times \sqrt{a} = a$, and leave the final answer in exact form unless told otherwise.
• 💡For a named term, write the power of the surd explicitly, for example $(\sqrt{2})^3$, then simplify it to $2\sqrt{2}$.
• 💡Check whether the question asks for an exact surd answer or a decimal approximation, and match the form requested.
• 💡Always write down the common difference first; it immediately gives the coefficient of n.
• 💡Substitute n = 1 into your expression and check it equals the first term before moving on.
• 💡If the sequence decreases, double-check that you have used a negative coefficient for n.
• 💡For multiple-choice questions, test each option with n = 1 and n = 2 rather than trying to spot the answer.
• 💡Write out the first and second differences clearly below the sequence to avoid arithmetic slips.
• 💡Subtract your an² terms carefully from the original sequence to reveal the linear pattern.
• 💡Check your final expression with n = 1 and n = 2 before selecting your final answer.
• 💡Always show your method for finding the nth term, especially for quadratic sequences. Even if your final answer is wrong, you can earn method marks for correctly finding differences or setting up equations.
• 💡When asked whether a number is in a sequence, set up an equation and solve it. If n is not a positive whole number, the number is not in the sequence. State this conclusion clearly.
• 💡For sequences with negative terms, use brackets when substituting into the nth term rule to avoid sign errors. For example, if n = -2, write (-2) squared, not -2 squared.

• Confusing term-to-term and position-to-term rules: for example, treating 'add 3' as if it meant the nth term is 3n. Correction: a term-to-term rule describes the step between consecutive terms, while a position-to-term rule gives a formula in n.
• Starting a position-to-term sequence at n = 0 instead of n = 1. Correction: the first term corresponds to n = 1, so substitute n = 1 first.
• Applying a term-to-term rule to the wrong starting value, such as using the second term as the starting point when the first term is given. Correction: always begin with the stated first term.
• Arithmetic slips when substituting into a position-to-term rule, for example calculating 4 × 3 − 1 as 8 by subtracting before multiplying, instead of the correct 11. Correction: substitute carefully and follow the correct order of operations.
• Mixing up square and cube numbers, for example thinking 8 is a square number. Correction: 8 is 2³, a cube number; 9 is 3², a square number.
• Confusing triangular numbers with square numbers, for example treating 1, 3, 6, 10 as squares. Correction: triangular numbers are sums of consecutive integers, while square numbers are n².
• Assuming a sequence is arithmetic when the differences are not constant. Correction: check that the difference between every pair of consecutive terms is the same before calling it an arithmetic progression.
• Forgetting that the first term of a sequence corresponds to n = 1 when using nth term formulas. Correction: substitute n = 1 for the first term, n = 2 for the second, and so on.
• Treating any sequence with a pattern as Fibonacci-type: the correction is that a Fibonacci-type sequence specifically adds the two previous terms, so 4, 7, 10, 13 is arithmetic, not Fibonacci-type.
• Assuming every quadratic sequence has second difference 1: the correction is that the second difference can be any non-zero constant, such as 2 or 6.
• Using a constant difference for a geometric progression: the correction is that a geometric progression has a constant multiplier r, so 2, 4, 8, 16 multiplies by 2 each time.
• Allowing r to be zero or negative when the row restricts r to a rational number greater than 0: the correction is to check that each term is obtained by multiplying by the same positive rational value.
• Treating $\sqrt{2}$ as $1.4$ and multiplying decimals: the correction is to keep the exact surd and simplify, since $\sqrt{2} \times \sqrt{2} = 2$ exactly.
• Adding the ratio instead of multiplying by it: the correction is that a geometric progression multiplies each term by $r$, so $1, \sqrt{2}, 2, 2\sqrt{2}$ multiplies by $\sqrt{2}$ each time.
• Writing $\sqrt{2} \times \sqrt{2}$ as $\sqrt{4}$ without simplifying: the correction is that $\sqrt{4} = 2$, so the product is the integer $2$.
• Assuming a surd ratio makes the sequence quadratic: the correction is that a constant multiplier, even an irrational one, defines a geometric progression.
• Writing the common difference as the constant term instead of the coefficient of n. The common difference belongs with n, so for 4, 7, 10, ... the expression is 3n + 1, not n + 3.
• Forgetting to include the constant adjustment after finding the coefficient. For 2, 5, 8, ... the expression is 3n - 1, not just 3n.
• Assuming the first term always equals the constant. The constant is found by comparing the first term with the value of an when n = 1, so it is not always the first term.
• Ignoring negative common differences. For 10, 7, 4, ... the common difference is -3, so the expression is -3n + 13, not 3n + something.
• Using the second difference directly as the coefficient of n² instead of halving it; the correction is to divide the second difference by 2 to find a.
• Forgetting to subtract the an² component from the original sequence before finding the linear part; the correction is to subtract an² from each term to find the linear sequence.
• Assuming the constant term c is simply the first term of the sequence; the correction is to derive it from the linear remainder sequence.
• Students often think that the common difference is always positive. Correct: if a sequence decreases, the common difference is negative, and the coefficient of n in the nth term will be negative.
• Students sometimes confuse the term number with the term value. Correct: the term number is the position (n), while the term value is the actual number in the sequence at that position.
• When finding the nth term of a quadratic sequence, students may forget to divide the second difference by 2 to get the coefficient of n squared. Correct: the coefficient of n squared is always half the second difference.

1. 1Day 1-2: Revise the basics of linear sequences. Practice finding the nth term of sequences that increase and decrease, and generate terms from a given rule.
2. 2Day 3-4: Learn how to identify and find the nth term of quadratic sequences. Focus on the method of differences and forming simultaneous equations.
3. 3Day 5-6: Study special sequences (Fibonacci, square, cube, triangular) and practice recognising them and finding specific terms.
4. 4Day 7-8: Work through exam-style questions on sequences, including problem-solving questions that ask whether a number belongs to a sequence.
5. 5Day 9-10: Complete a timed practice paper or set of mixed questions, and review any mistakes to ensure you understand the correct methods.

• 📋Finding the nth term of a linear sequence: You will be given a sequence and asked to find an expression for the nth term. Always check your answer by substituting n = 1, 2, and 3.
• 📋Finding the nth term of a quadratic sequence: You will be given a sequence with a constant second difference. Show your working for the differences and use simultaneous equations to find the full expression.
• 📋Determining whether a number is a term of a sequence: Set the nth term equal to the given number and solve for n. If n is a positive integer, the number is in the sequence; otherwise, it is not. Always state your conclusion.
• 📋Generating terms from a given nth term rule: Substitute n = 1, 2, 3, etc., into the rule. Be careful with negative signs and brackets.