Sequences

    OCR
    GCSE

    Sequences are ordered sets defined by specific algebraic rules, necessitating fluency in both term-to-term and position-to-term definitions. Analysis requires the derivation of the $n$th term for linear ($an+b$), quadratic ($an^2+bn+c$), and geometric ($ar^{n-1}$) progressions, alongside recursive definitions such as the Fibonacci sequence. Mastery involves investigating limiting behaviours as $n \to \infty$, manipulating sigma notation for summations, and constructing algebraic proofs for sequence properties. This topic bridges discrete arithmetic patterns with continuous algebraic functions, establishing the foundational logic for series and calculus.

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    Objectives
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    Exam Tips
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    Pitfalls
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    Key Terms
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    Mark Points

    What You Need to Demonstrate

    Key skills and knowledge for this topic

    • Award M1 for correctly determining the second difference to establish the coefficient of n² in a quadratic sequence
    • Award A1 for the fully simplified nth term expression in the form an² + bn + c
    • Credit responses that correctly set up simultaneous equations to determine the first term (a) and common ratio (r) in geometric progressions
    • Award B1 for explicitly stating the limiting value of a sequence as n approaches infinity, not just calculating a large term
    • Award 1 mark for rejecting non-integer solutions for n when determining if a specific value exists within a sequence

    Example Examiner Feedback

    Real feedback patterns examiners use when marking

    • "You correctly identified the second difference. Now, ensure you halve this value to find the coefficient of n²"
    • "Your nth term formula is correct, but you must show the check step (substituting n=1, 2) to guarantee full marks"
    • "You found a value for n, but is it an integer? Remember, sequence positions must be whole numbers"
    • "Excellent work on the geometric ratio. To reach the top band, explain clearly why the sequence converges to a limit"

    Marking Points

    Key points examiners look for in your answers

    • Award M1 for correctly determining the second difference to establish the coefficient of n² in a quadratic sequence
    • Award A1 for the fully simplified nth term expression in the form an² + bn + c
    • Credit responses that correctly set up simultaneous equations to determine the first term (a) and common ratio (r) in geometric progressions
    • Award B1 for explicitly stating the limiting value of a sequence as n approaches infinity, not just calculating a large term
    • Award 1 mark for rejecting non-integer solutions for n when determining if a specific value exists within a sequence

    Examiner Tips

    Expert advice for maximising your marks

    • 💡Always verify your calculated nth term rule by substituting n=1, 2, and 3 to ensure it generates the original sequence
    • 💡When asked for a limiting value, you must write 'as n increases' or 'as n → ∞' to gain the communication mark
    • 💡For quadratic sequences, use the 'compare coefficients' method or 'sequence subtraction' method rather than trial and error
    • 💡In geometric sequence questions, look for the keyword 'exact'—this is a command to leave your answer in surd or fraction form

    Common Mistakes

    Pitfalls to avoid in your exam answers

    • Halving the second difference incorrectly or forgetting to halve it when finding the coefficient of n²
    • Confusing the position of the term (n) with the value of the term (u_n) when solving algebraic problems
    • Assuming a sequence is geometric by checking only the first two terms without verifying the common ratio across subsequent terms
    • Rounding values too early when working with geometric sequences involving surds or fractions

    Key Terminology

    Essential terms to know

    Derivation of $n$th term formulae (Linear, Quadratic, Geometric)
    Recursive definitions and term-to-term rules
    Limiting values and convergence of sequences
    Sigma notation and summation of finite series

    Likely Command Words

    How questions on this topic are typically asked

    Find
    Determine
    Show that
    Calculate
    Write down

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