Lesson 8.01e-i

8.01e-i Fibonacci numbers, solving recurrences and modelling Quiz: OCR Further Maths, Unit 5

20 questions

Written by Revision Ninja · Last reviewed 11 October 2026

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Lesson 8.01e-i, Fibonacci numbers, solving recurrences and modelling: 20 multiple choice questions for the OCR Further Maths (H245), Unit 5: Additional Pure Mathematics (Y545), written with Revision Ninja.

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The 20 questions

  1. Which recurrence relation defines the standard Fibonacci sequence for n >= 2?

    • F_n = 2 F_{n-1}
    • F_n = F_{n-1} F_{n-2}
    • F_n = F_{n-1} + F_{n-2}
    • F_n = F_{n-1} - F_{n-2}

  2. What is the base value phi in Binet's formula for Fibonacci numbers?

    • (1 + sqrt(5)) / 2
    • (1 + sqrt(3)) / 2
    • (1 - sqrt(5)) / 2
    • sqrt(5) / 2

  3. What is the order of the recurrence relation u_n = 3u_{n-1} - 2u_{n-2} + 5?

    • First order
    • Second order
    • Fifth order
    • Third order

  4. What is the characteristic equation of the recurrence relation u_n = 5u_{n-1} - 6u_{n-2}?

    • r^2 - 6r + 5 = 0
    • r^2 - 5r + 6 = 0
    • r^2 + 6r - 5 = 0
    • r^2 + 5r - 6 = 0

  5. What are the characteristic roots for the recurrence relation u_n = 4u_{n-1} - 4u_{n-2}?

    • r = 4 repeated
    • r = 4 and 1
    • r = 2 repeated
    • r = 2 and -2

  6. What is the general solution of a linear recurrence with distinct characteristic roots r_1 and r_2?

    • u_n = A r_1^n + B r_2^n
    • u_n = A r_1 + B r_2
    • u_n = (A + Bn) r_1^n
    • u_n = A r_1^n B r_2^n

  7. What is the general solution for a second-order recurrence with a repeated characteristic root r?

    • u_n = (A + Bn) r^n
    • u_n = A r^n + B r^n
    • u_n = A r^n + B n^r
    • u_n = A n r^n

  8. What is the solution to u_n = 3u_{n-1} given u_0 = 2?

    • u_n = 2 x 3^n
    • u_n = 3 x 2^n
    • u_n = 2 + 3^n
    • u_n = 6^n

  9. Taking F_0 = 0 and F_1 = 1, what is the value of the Fibonacci number F_6?

    • 6
    • 13
    • 5
    • 8

  10. What is the limit of F_{n+1} / F_n as n approaches infinity?

    • Infinity
    • Silver ratio
    • Euler's number
    • Golden ratio

  11. What form of particular solution should be tried for u_n - 3u_{n-1} = 4?

    • C 3^n
    • C n^2
    • C n
    • Constant C

  12. If the characteristic equation root is 2 and non-homogeneous term is 3 x 2^n, try:

    • C n
    • C n^2 2^n
    • C n 2^n
    • C 2^n

  13. Which power series represents the generating function G(x) for a sequence a_n?

    • sum a_n x^n
    • sum (a_n + x)^n
    • sum a_n n^x
    • sum a_n / x^n

  14. What is the closed-form generating function for the constant sequence a_n = 1?

    • 1 / (1 - x)
    • 1 / (1 + x)
    • x / (1 - x)
    • 1 / x

  15. What is the closed-form generating function for Fibonacci numbers with F_0 = 0, F_1 = 1?

    • x / (1 + x + x^2)
    • 1 / (1 - x - x^2)
    • 1 / (1 - 2x)
    • x / (1 - x - x^2)

  16. What is the formula for the sum of the first n Fibonacci numbers F_1 + ... + F_n?

    • F_{n+1}
    • F_{n+1} - 1
    • F_{n+2} - 1
    • F_{n+2} + 1

  17. What is the simplified value of Cassini's identity F_{n-1} F_{n+1} - F_n^2?

    • (-1)^{n+1}
    • 1
    • (-1)^n
    • 0

  18. What component makes a linear recurrence relation non-homogeneous?

    • Non-zero extra term
    • Variable coefficients
    • Negative coefficients
    • Repeated characteristic roots

  19. What is the solution to u_n = u_{n-1} + 2 given u_0 = 1?

    • u_n = n + 2
    • u_n = 2^n
    • u_n = 2n - 1
    • u_n = 2n + 1

  20. A population doubles each year and 3 die. Which recurrence relation models this?

    • P_n = P_{n-1}^2 - 3
    • P_n = 2 P_{n-1} - 3
    • P_n = 2 P_{n-1} + 3
    • P_n = 3 P_{n-1} - 2

All OCR Further Maths quizzes