Sequences — AQA GCSE Mathematics
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Sequences explained
A term-to-term rule tells you how to get from one term to the next, so you need a starting value and then you apply the rule again and again.
Read the full explanation
If the first term is 4 and the rule is "multiply by 3 then subtract 2", the sequence runs 4, 10, 28, 82. A position-to-term rule, usually written as an nth term expression, works out any term straight from its position: with nth term 5n − 3, the 20th term is 5 × 20 − 3 = 97. Questions give you a rule in words or in symbols and ask for the first few terms, or for a term a long way down the list. When the position is large, the position-to-term rule is far quicker, because a term-to-term rule forces you to grind through every term in between.
recognise and use sequences of triangular, square and cube numbers and simple arithmetic progressions
Triangular numbers count dots stacked in a triangle: 1, 3, 6, 10, 15, where each step adds one more than the step before. Square numbers are 1, 4, 9, 16, 25, coming from 1², 2², 3² and so on, and cube numbers are 1, 8, 27, 64, 125, from 1³, 2³, 3³. An arithmetic progression rises or falls by the same amount every time, such as 7, 11, 15, 19 with a common difference of 4. Learn the squares up to 15² and the cubes up to 5³ by heart, because questions hide them inside puzzles, for example asking which value in a list is both square and triangular. You may also be shown a growing pattern of shapes and asked how many tiles the next diagram needs.
including Fibonacci-type sequences, quadratic sequences, and simple geometric progressions (rⁿ where n is an integer and r is a rational number > 0)
This topic covers sequences beyond linear ones. A Fibonacci-type sequence adds the two previous terms to get the next; in 2, 5, 7, 12, ..., the next term is 19. A quadratic sequence has a constant second difference; for 3, 8, 15, 24, ..., the first differences are 5, 7, 9, and the second differences are a constant 2. A geometric progression (GP) has a common ratio (multiplier) between terms. Find the ratio by dividing a term by the one before it. The specification highlights the simple GP form rⁿ, where n is an integer and r is a rational number > 0. For example, if r = 4, the sequence is 4¹, 4², 4³, ... which is 4, 16, 64, ...
including other sequences including where r is a surd (Higher tier only)
This is Higher tier only. Some geometric sequences have a common ratio that is a surd, so when the first term is a whole number the terms alternate between surd form and whole numbers. Take 1, √3, 3, 3√3, 9: each term is the one before it multiplied by √3, because √3 × √3 = 3. To find the ratio, divide a term by the previous one and rationalise if you need to, for example 3 ÷ √3 becomes √3 once you multiply top and bottom by √3. You may also meet sequences built from other rules, including ones that mix a linear part with a multiplying part. Work in exact surd form the whole way through instead of switching to decimals, since exact answers are wanted, and remember that squaring a surd removes the root sign, so (√5)² = 5.
deduce expressions to calculate the nth term of linear sequences
A linear sequence rises or falls by the same amount each time, and that common difference becomes the number in front of n. For 7, 10, 13, 16 the difference is 3, so start with 3n, which gives 3, 6, 9, 12. Every term of the sequence is 4 more than the matching value of 3n, so the rule is 3n + 4. Check it by putting the first position back in: 3 × 1 + 4 = 7. A falling sequence gives a negative multiplier, so 20, 17, 14 leads to 23 − 3n. Once you have the rule you can answer "is 85 a term?" by solving 3n + 4 = 85, which gives n = 27; a whole number position of 1 or more means yes, and a position that is not a whole number, or is less than 1, means no.
including quadratic sequences (Higher tier only)
This is Higher tier only. A quadratic sequence has a constant second difference, and the number in front of n² is half of that second difference. Take 4, 11, 22, 37, 56. The first differences are 7, 11, 15, 19 and the second difference is 4, so the squared part is 2n². Write out 2n² for the first few positions: 2, 8, 18, 32, 50. Take those away from the original terms and you are left with 2, 3, 4, 5, 6, a linear sequence whose rule is n + 1. Putting the two parts together gives 2n² + n + 1. Always check by substituting a position back in: the third term should be 22, and 2 × 9 + 3 + 1 = 22, so the rule works.
Your focus
- Describe how a term-to-term rule needs a starting value while a position-to-term rule works from the position alone.
- Work out a distant term by substituting its position number into an nth term rule such as 5n − 3.
- Explain why a position-to-term rule is far quicker than a term-to-term rule for a term a long way down the list.
Show all 20 objectives
- Write down the triangular, square and cube numbers from memory, squares to 15² and cubes to 5³.
- Work out how many tiles the next diagram of a growing pattern needs by showing how many are added each time.
- Show that a given value does or does not belong to a sequence, supporting the answer with a reason.
- Describe the rule for each family: adding the two previous terms, a constant second difference, or a constant multiplying ratio.
- Find a common ratio by dividing a term by the one before it and check the same ratio works for other consecutive pairs.
- Work out a missing term of a Fibonacci-type sequence by working backwards (subtracting) as well as forwards (adding).
- Show that a sequence is quadratic by writing the first differences and finding a constant second difference.
- Use the form rⁿ, with n an integer and r a rational number greater than 0, to generate terms of a simple geometric progression.
- Explain why a ratio of √3 makes terms alternate between surd form and whole numbers, since √3 × √3 = 3.
- Work out the common ratio by dividing consecutive terms and rationalising a denominator where a root sits underneath.
- Find the next term in exact simplified surd form rather than switching to a rounded decimal.
- Explain why the common difference becomes the number in front of n in the nth term expression.
- Find the nth term of a linear sequence by writing the multiple of n first and then adjusting by a constant.
- Show that a value is or is not a term by solving for n and testing whether n is a whole number of 1 or more.
- Explain why the coefficient of n² is half of the constant second difference.
- Find the nth term of a quadratic sequence by subtracting the squared part to leave a linear sequence to solve.
- Show that a rule is correct by substituting a position back in and matching the term it produces.
Sequences exam tips
Quick Revision Summary (Key Takeaway)
A sequence is an ordered list of numbers generated by a specific rule, where each number is called a term. At AQA GCSE, you must find the nth term of linear and quadratic sequences, generate terms from a given rule, and recognise special sequences such as Fibonacci, square and triangular numbers.
Topic Overview
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.
Key Concepts
- →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.
Marking Points
- substituting the position number into a position-to-term rule, for example writing 5 × 20 − 3, even if the arithmetic that follows is wrong
- showing a term-to-term rule applied at least once, so the examiner can see it was applied to the previous term and not to the position number
- the terms themselves, correct and in the right order, with any negatives or decimals written accurately
- using the correct starting value when applying a term-to-term rule, so the first generated term follows from the given term
- naming the type of sequence, for example stating that the values are triangular or that the common difference is 4
- the correct next term or terms of the pattern, even if a later part of the question goes wrong
- a method taken from a diagram, such as showing how many extra tiles are added each time
- a supported reason when you are asked whether a given value belongs to the sequence
- Stating the rule that governs the sequence, such as the common ratio or the constant second difference, even if a later term is wrong.
- A correct row of first differences when a quadratic sequence is being tested.
- Showing the addition or multiplication used, since a bare list of terms can lose the method mark.
- Using rⁿ with n as an integer and r as a rational number > 0 to find a term of a simple geometric progression.
- a common ratio left in exact surd form, such as √2 rather than a rounded decimal
- correct use of a surd rule, for example √3 × √3 = 3, even if the final term is wrong
- the next term written exactly, with the surd fully simplified
- rationalising a denominator when the ratio comes out as a fraction with a root underneath
- a correct common difference, for example writing 3 or saying the sequence goes up in threes
- the correct multiple of n, such as 3n, before the constant has been worked out
- the complete expression written in terms of n rather than as a list of terms
- setting up and solving an equation when you are asked to decide whether a value is a term
- a correct row of first differences and a correct constant second difference
- halving the second difference to get the squared part, for example 2n²
- subtracting the squared part from the original terms to leave a linear sequence
- the complete expression in terms of n, with the linear part correctly attached
Examiner Tips
- 💡Write the terms in a row with the position numbers above them, so you can see at a glance which kind of rule you have been given.
- 💡When a question asks for a term far along the sequence, look for an nth term expression rather than listing every term up to it.
- 💡Substitute your answer back into the rule as a check before moving on.
- 💡Write the differences between terms underneath the sequence; a constant difference confirms an arithmetic progression.
- 💡Keep the squares and cubes at your fingertips, since they turn up in surds, area and Pythagoras questions as well as in sequences.
- 💡With a pattern of shapes, count the shapes in the first three diagrams before you look for a rule.
- 💡Set out two rows of differences under the sequence before you decide what type it is.
- 💡For a geometric progression, work out the ratio twice from different pairs of terms to confirm it is constant.
- 💡A ratio between 0 and 1 makes the terms decrease, so do not assume a geometric progression always grows.
- 💡Keep every line in surd form and only tidy the surd at the end.
- 💡Use the fact that squaring a surd removes the root, which turns a surd ratio into a whole number quickly.
- 💡Write the ratio as a fraction first and rationalise it, rather than guessing it from the size of the terms.
- 💡Write the multiples of the common difference above the sequence and compare row by row; the constant is whatever you must add to get from one row to the other.
- 💡Test your expression on the second or third term as well as the first, since a rule that fits only the first term can still be wrong.
- 💡To decide whether a value belongs to the sequence, set your expression equal to it and solve; a position that is not a whole number of 1 or more means it is not a term.
- 💡Lay the sequence, the first differences and the second differences out in three rows before you write anything else.
- 💡After subtracting the squared part, what is left must be linear; if it is not, go back and check your differences.
- 💡Substitute the second position into your final expression as a quick check against the second term.
- 💡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.
Common Mistakes
- using the position number inside a term-to-term rule, so the fifth term is worked out from 5 rather than from the fourth term
- starting a term-to-term rule in the wrong place, for example treating the given first term as though it were term zero
- reading an expression such as 5n − 3 as subtract first then multiply, instead of multiplying by the position number first
- muddling square and triangular numbers, so writing 1, 4, 9, 16 when the question wants 1, 3, 6, 10
- cubing by multiplying by 3, giving 12 for 4³ instead of 64
- calling any sequence with a growing gap arithmetic, when an arithmetic progression needs the same difference every time
- Treating a geometric progression as arithmetic, for example adding 3 each time instead of multiplying by 3.
- Deciding a sequence has no rule because the first differences change, instead of going on to work out the second differences.
- In a Fibonacci-type sequence, adding the wrong pair of terms, for example the first and third, instead of the two immediately preceding terms.
- rounding the ratio to a decimal early, which throws away the exact answer the question asks for
- working out √3 × √3 as 9, by squaring the number under the root sign as well as removing the sign
- adding the ratio instead of multiplying, so writing 2√3 after √3 rather than 3
- giving a term-to-term rule such as "add 3" when an expression in n is what the question wants
- using the first term as the constant, so writing 3n + 7 for 7, 10, 13 instead of 3n + 4
- losing the sign on a falling sequence, writing 3n − 23 rather than 23 − 3n
- using the whole second difference as the multiplier, so writing 4n² when the second difference is 4
- jumping straight to guessing the constant without subtracting the squared part first
- lining the differences up with the wrong positions, which shifts the linear part by one term
- 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.
Revision Plan
- 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.
- 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.
- 3Day 5-6: Study special sequences (Fibonacci, square, cube, triangular) and practice recognising them and finding specific terms.
- 4Day 7-8: Work through exam-style questions on sequences, including problem-solving questions that ask whether a number belongs to a sequence.
- 5Day 9-10: Complete a timed practice paper or set of mixed questions, and review any mistakes to ensure you understand the correct methods.
Exam Question Types
- 📋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.
Command Word Expectations (AQA)
In AQA GCSE Mathematics, 'Find' requires you to calculate a value or expression. You must show sufficient working to make your method clear, especially for multi-step problems. For example, 'Find the nth term' means you must derive the expression, not just guess it.
'Show that' means you must provide a logical chain of reasoning that leads to the given result. You must include every step of your working, and the final line should match the statement you are asked to show. For example, 'Show that the nth term is 3n + 2' requires you to demonstrate how you arrived at that expression.
'Explain' requires you to give a reason or justification for your answer, often in words. For example, 'Explain why 100 is not a term of the sequence' means you must show that solving the nth term equation does not give a positive integer, and state that this is why 100 is not in the sequence.
How Students Lose Marks (Examiner Pitfalls)
Step-by-Step Worked Solutions
Question: The nth term of a sequence is given by 4n - 7. (a) Find the first three terms of the sequence. (b) Which term of the sequence is equal to 45?
- 1.Step 1: Substitute n = 1 into the rule 4n - 7 to find the first term: 4(1) - 7 = -3.
- 2.Step 2: Substitute n = 2 to find the second term: 4(2) - 7 = 1.
- 3.Step 3: Substitute n = 3 to find the third term: 4(3) - 7 = 5.
- 4.Step 4: For part (b), set the nth term equal to 45: 4n - 7 = 45.
- 5.Step 5: Solve for n: add 7 to both sides to get 4n = 52, then divide by 4 to get n = 13.
- 6.Step 6: State the final answer: the first three terms are -3, 1, 5 and the 13th term is 45.
Question: A quadratic sequence begins 5, 12, 23, 38, 57, ... Find an expression for the nth term of this sequence.
- 1.Step 1: Find the first differences: 12 - 5 = 7, 23 - 12 = 11, 38 - 23 = 15, 57 - 38 = 19.
- 2.Step 2: Find the second differences: 11 - 7 = 4, 15 - 11 = 4, 19 - 15 = 4. The second difference is constant at 4, so the sequence is quadratic.
- 3.Step 3: The coefficient of n squared is half the second difference: 4 divided by 2 = 2. So the nth term starts with 2n squared.
- 4.Step 4: Subtract 2n squared from each term to find the remaining linear sequence. For n = 1: 5 - 2(1) = 3. For n = 2: 12 - 2(4) = 4. For n = 3: 23 - 2(9) = 5. For n = 4: 38 - 2(16) = 6. The remaining sequence is 3, 4, 5, 6, ... which has nth term n + 2.
- 5.Step 5: Combine to get the full nth term: 2n squared + n + 2.
- 6.Step 6: Check with n = 5: 2(25) + 5 + 2 = 50 + 7 = 57, which matches the fifth term.