Algebra - Sequences — Edexcel GCSE Mathematics

A sequence is an ordered list of numbers generated by a term-to-term or position-to-term rule. At GCSE you must find the nth term of linear and quadratic sequences, recognise special sequences (triangular, square, Fibonacci), and use sequences to solve problems involving patterns and diagrams.

Sequences are ordered lists of numbers that follow a specific pattern or rule. In GCSE Mathematics, you will encounter linear sequences (constant difference), quadratic sequences (constant second difference), and special sequences such as Fibonacci, triangular, and square numbers. You must be able to find the nth term of a sequence, determine whether a given number belongs to a sequence, and generate terms from a given rule.

This topic is fundamental to algebra and problem-solving, appearing in both calculator and non-calculator papers. It links to functions, graphs, and later topics like calculus. Mastering sequences builds logical reasoning and pattern recognition, which are essential for higher-tier questions and real-world applications such as predicting trends and modelling growth.

• →A sequence is an ordered list of terms, often denoted u1, u2, u3, ... or T(n).
• →Linear sequences have a constant first difference and nth term of the form an + b, where a is the common difference and b is found by substituting n=1.
• →Quadratic sequences have a constant second difference; the coefficient of n^2 is half the second difference, and the remaining linear part is found by subtracting n^2 terms.
• →Special sequences include Fibonacci (each term is the sum of the two previous), triangular numbers (n(n+1)/2), and square numbers (n^2).
• →The nth term formula allows you to find any term in a sequence without listing all previous terms, and to test whether a given number is a term.

• applying a term-to-term rule given in words to the starting value at least once, since every later term is built on that step
• substituting the position numbers into a position-to-term rule, even if one of the terms is then evaluated wrongly
• the list of terms in the right order, with as many terms as the question asked for
• the single term asked for, such as the 10th term, when the question skips the list
• Correctly identifying the type of sequence with a supporting reason, for example, 'quadratic because the second differences are constant'.
• Calculating the common difference of an arithmetic progression or the common ratio of a geometric progression.
• Finding the next term(s) in a sequence by correctly applying its rule.
• Providing a clear reason why a given number is or is not in a sequence, for example by showing all terms in the sequence are odd but the given number is even.
• the common ratio, including when it is a surd, such as dividing 2√2 by 2 to get √2
• a term left in exact surd form rather than rounded to a decimal
• stating the rule of an unfamiliar sequence, in words or in algebra, even if a later term is wrong
• testing a proposed rule against a term you were given, which supports the rule but does not by itself prove it holds for every term
• recognising and continuing the standard families, including triangular, square, cube, arithmetic, Fibonacci and quadratic sequences
• the common difference identified, and the first part of the rule written as that difference multiplied by n
• the constant found by comparing the multiples of the difference with the given terms
• the complete expression in n, such as 4n − 1, rather than a term-to-term rule written in words
• substituting a position back into the expression to justify an answer when asked to show that a number is or is not a term
• for a membership question, solving the equation and interpreting the solution: a positive whole number means the value is a term, otherwise it is not
• for a linear sequence, the common difference identified and used as the coefficient of n
• the first and second differences set out under the sequence when testing for a quadratic
• the coefficient of n², taken as half of the constant second difference
• subtracting the multiples of n² from the terms and finding the rule of what is left, even if the final expression is wrong
• the complete expression in n, such as 2n² − n + 3, with the squared part and the linear part combined

• 💡Write the positions 1, 2, 3, 4 in a row with the terms underneath, so you can see which term belongs to which position.
• 💡If a question asks for a term far along, such as the 100th, use a position-to-term rule rather than counting on step by step.
• 💡Check your list by putting the last position back into the rule; if it does not return your last term, the slip is in the list.
• 💡Always start by finding the differences between terms. If the first differences are constant, it's arithmetic. If the second differences are constant, it's quadratic. If differences don't reveal a pattern, test for a common ratio (geometric) or the Fibonacci rule.
• 💡Memorise the first few terms of special sequences to save time: triangular numbers up to 36 (1, 3, 6...), square numbers up to 144 (12²), and cube numbers up to 125 (5³).
• 💡Keep surds exact the whole way through; a term such as 8√3 is the form the mark scheme wants, not a rounded 13.9.
• 💡If neither the differences nor the ratios are constant, look at the terms as fractions or as powers before you decide there is no rule.
• 💡Say which family a sequence belongs to in words before you calculate, because the reason often carries a mark of its own.
• 💡Check the standard families first: if the gaps are constant it is arithmetic, if the second differences are constant it is quadratic, and if the ratios are constant it is geometric.
• 💡Write the difference above each gap first; if the differences are not all the same, the sequence is not linear and this method will not give a valid rule.
• 💡Test your expression on the last term you were given, not the first, because that catches an error in the difference as well as in the constant.
• 💡Leave the answer as an expression such as 6n − 4; attaching an equals sign and a number answers a different question.
• 💡Set out three rows, the terms, the first differences and then the second differences, so the constant row is visible before you commit to a method.
• 💡Check the finished expression at two different positions, one of them the last term you were given, because a wrong constant can still match a single term by chance.
• 💡If the first differences are already constant, the sequence is linear and the squared step is not needed at all.
• 💡Always show your method for finding the nth term, especially for quadratic sequences. Writing out first and second differences can earn method marks even if the final answer is wrong.
• 💡When asked if a number is in a sequence, set up an equation and solve for n. State clearly that n must be a positive integer, and conclude accordingly.
• 💡For term-to-term rules, write out each step clearly. If asked for a large term (e.g., 10th term), consider whether a formula would be quicker, but only if you are confident in deriving it.

• starting a position-to-term rule at the position zero, so 3n + 2 is written as beginning at 2 rather than at 5
• reading a position-to-term rule as a term-to-term one, so treating 3n + 2 as an instruction to triple the previous term and add two
• applying a two-step term-to-term rule in the wrong order, since 'double then subtract 3' and 'subtract 3 then double' build different sequences
• writing the terms of a term-to-term rule without using the given starting value, which shifts the whole list
• Mistaking a quadratic sequence for an arithmetic one. For example, in the square numbers 1, 4, 9, 16, the first differences are 3, 5, 7. This is a pattern, but not a constant difference. Correction: Always check the second differences; if these are constant, the sequence is quadratic.
• Confusing cube numbers (n³) with multiples of three (3n), for example writing 3, 6, 9 instead of 1³, 2³, 3³ which are 1, 8, 27. Correction: Remember that 'cubed' means a number multiplied by itself twice (e.g., 5³ = 5 × 5 × 5 = 125).
• In a Fibonacci-type sequence, adding the wrong terms to find the next one, for example in 2, 5, 7, 12, ..., adding 2 + 12 instead of 7 + 12. Correction: The rule is to always add the two most recent terms to get the next term (e.g., 7 + 12 = 19).
• rounding √2 to 1.41 part way through, so the later terms drift and no longer match the sequence
• squaring the ratio when only one step is taken, so multiplying by 2 where the step is a multiplication by √2
• assuming an unfamiliar sequence must be quadratic and forcing second differences that are not actually constant
• treating a sequence of fractions as though only the numerators matter, and missing that the denominators follow their own rule
• using the first term as the constant, so writing 3n + 5 for 5, 8, 11 when the rule is 3n + 2; correct by comparing 3n with the terms
• answering with a term-to-term instruction such as 'add 3 each time' when an expression in n was asked for; correct by giving the rule in terms of n
• losing the sign on a decreasing sequence, giving 3n + 23 rather than 23 − 3n; correct by checking position one
• answering with the value of a particular term instead of the rule, so writing 17 when the question wanted the expression; correct by writing the expression in n
• using the second difference itself as the coefficient, so writing 4n² when the second difference is 4 and the coefficient is 2; correct by halving the second difference
• subtracting the terms from the squares instead of the squares from the terms, which reverses the sign of everything that is left; correct by subtracting squares from terms
• stopping at the squared part and leaving out the linear sequence that remains; correct by finding the rule of the leftovers and adding it
• deciding a sequence is quadratic after checking only one pair of second differences; correct by checking that all second differences are equal
• Students often think the nth term is the same as the term-to-term rule. For example, for 2, 5, 8, 11, ... the term-to-term rule is 'add 3', but the nth term is 3n - 1. The nth term gives the value of any term directly from its position n.
• When finding the nth term of a quadratic sequence, students may forget to halve the second difference or may not subtract the n^2 part correctly. Always check by substituting n=1, 2, 3 into your final formula.
• Students sometimes assume a sequence must be arithmetic or geometric. Sequences can follow any rule, such as multiplying by 2 and adding 1, or alternating operations.

1. 1Day 1-2: Revise linear sequences. Practice finding the nth term of arithmetic sequences and using it to find specific terms. Complete at least 10 practice questions.
2. 2Day 3-4: Move on to quadratic sequences. Learn the method of first and second differences. Practice finding nth term for at least 5 quadratic sequences, checking each answer.
3. 3Day 5-6: Study special sequences (Fibonacci, triangular, square) and term-to-term rules. Practice generating terms and identifying patterns.
4. 4Day 7-8: Work through exam-style questions on sequences, including problem-solving and reasoning questions. Focus on showing clear method.
5. 5Day 9-10: Review common misconceptions and examiner tips. Complete a timed practice paper section on sequences and mark it using the mark scheme.

• 📋Find the nth term of a linear sequence (2-3 marks). Advice: Identify the common difference, then find the constant term by substituting n=1.
• 📋Find the nth term of a quadratic sequence (3-4 marks). Advice: Show first and second differences, halve the second difference, then subtract n^2 to find the linear part.
• 📋Determine whether a given number is a term in a sequence (2-3 marks). Advice: Set the nth term equal to the number, solve for n, and check if n is a positive integer.
• 📋Generate terms from a term-to-term rule or solve a problem involving a sequence pattern (3-5 marks). Advice: Write out each step clearly and look for a pattern to find a specific term.