Sequences

OCR

GCSE

Sequences function as discrete mappings from the set of positive integers to real values, requiring candidates to distinguish rigorously between term-to-term (recursive) and position-to-term (algebraic) definitions. Mastery involves not only generating terms but deducing the $n$th term for linear, quadratic, and geometric progressions through analysis of first and second differences or common ratios. Assessment prioritizes the algebraic verification of term membership (AO2) and the synthesis of sequence properties to solve problems involving growth and decay (AO3).

Objectives

Exam Tips

Pitfalls

Key Terms

Mark Points

Subtopics in this area

Sequences

What You Need to Demonstrate

Key skills and knowledge for this topic

Award 1 mark for correctly identifying the common difference in a linear sequence
Award 1 mark for writing the correct term containing n (e.g., 4n) based on the common difference
Award 1 mark for the correct adjustment of the constant term to complete the linear nth term expression
Award method marks for calculating the second difference to determine the coefficient of n² in quadratic sequences (Higher only)
Credit responses that set the nth term expression equal to a value and solve for n to prove term membership
Award 1 mark for correctly identifying the common difference or common ratio between consecutive terms
Award 1 mark for the correct algebraic expression for the nth term (e.g., 3n - 2), specifically crediting the correct coefficient of n
For quadratic sequences (Higher), award method marks for calculating the second difference and halving it to determine the coefficient of n²

Example Examiner Feedback

Real feedback patterns examiners use when marking

"You correctly identified the common difference, but your nth term formula is incorrect—remember the difference becomes the coefficient of n"
"Excellent work finding the quadratic rule; ensure you show the subtraction step clearly to secure method marks if the final answer is wrong"
"You stated the number isn't in the sequence, but you must show the equation solving for n to prove it is not an integer"
"Good recognition of the geometric progression; now try to express the nth term using index notation"

Marking Points

Key points examiners look for in your answers

Award 1 mark for correctly identifying the common difference in a linear sequence
Award 1 mark for writing the correct term containing n (e.g., 4n) based on the common difference
Award 1 mark for the correct adjustment of the constant term to complete the linear nth term expression
Award method marks for calculating the second difference to determine the coefficient of n² in quadratic sequences (Higher only)
Credit responses that set the nth term expression equal to a value and solve for n to prove term membership
Award 1 mark for correctly identifying the common difference or common ratio between consecutive terms
Award 1 mark for the correct algebraic expression for the nth term (e.g., 3n - 2), specifically crediting the correct coefficient of n
For quadratic sequences (Higher), award method marks for calculating the second difference and halving it to determine the coefficient of n²
Credit responses that set the nth term expression equal to a given value and solve for n to prove term membership
Award 1 mark for correctly generating the first three terms when given a specific position-to-term rule

Examiner Tips

Expert advice for maximising your marks

💡Always verify your nth term rule by substituting n=1, n=2, and n=3 to ensure it generates the original sequence
💡When asked if a number is in a sequence, set your nth term equal to the number and solve; if n is not a whole number, state 'No' with the non-integer value as evidence
💡For Fibonacci-type sequences, write out the addition rule explicitly (e.g., term 3 = term 1 + term 2) to avoid calculation errors
💡Higher Tier: Memorize the difference method for quadratics (2a = 2nd diff) as it is faster than simultaneous equations
💡Always verify your calculated nth term by substituting n=1, n=2, and n=3 to check if it generates the original sequence correctly
💡When asked 'Is X a term in the sequence?', set your nth term rule equal to X and solve for n; conclude 'yes' only if n is a positive integer
💡For quadratic sequences, structure your working clearly: find the second difference, determine an², subtract this from the original sequence, and find the linear rule for the remainder
💡Memorize the structure of special sequences like Fibonacci and triangular numbers, as these are often tested without explicit formulae provided

Common Mistakes

Pitfalls to avoid in your exam answers

Writing 'n + 3' instead of '3n' for a sequence that increases by 3 (confusing addition rule with multiplication coefficient)
Confusing the term value with the position number (n) when substituting
Failing to identify that a sequence is geometric and attempting to find a linear common difference
In quadratic sequences, correctly finding 'a' (the n² coefficient) but making arithmetic errors when subtracting an² from the original sequence to find 'bn + c'
Confusing the term-to-term rule (e.g., 'add 3') with the position-to-term rule, often writing 'n + 3' instead of '3n' for a linear sequence
Failing to halve the second difference when finding the coefficient of n² in quadratic sequences, leading to an incorrect leading term
Assuming a sequence is arithmetic when it is actually geometric or Fibonacci-type, particularly when only the first three terms are analyzed
Stating that a number is not in a sequence without showing the calculation where n results in a non-integer

Key Terminology

Essential terms to know

Linear (Arithmetic) Sequences and nth Term

Quadratic Sequences and Second Differences

Geometric Progressions and Common Ratios

Fibonacci and Recurrence Relations

Linear (Arithmetic) Sequences and the nth term

Quadratic and Geometric Progressions

Special Sequences (Fibonacci, Triangular, Square, Cube)

Term-to-term vs Position-to-term rules

Likely Command Words

How questions on this topic are typically asked

Find

Write

Generate

Show that

Determine

Write down

Explain

Ready to test yourself?

Practice questions tailored to this topic