8.01a-c Recurrence relations and properties of sequences Quiz: OCR Further Maths, Unit 5

The 20 questions

1. What is the auxiliary equation for the recurrence relation u_{n+2} - 5u_{n+1} + 6u_n = 0?

   • λ² - 5λ + 6 = 0
   • λ² + 5λ + 6 = 0
   • λ² - 5λ - 6 = 0
   • 6λ² - 5λ + 1 = 0

2. What is the general solution if an auxiliary equation has distinct real roots α and β?

   • (A + Bn) αⁿ
   • A αⁿ βⁿ
   • A αⁿ + B n βⁿ
   • A αⁿ + B βⁿ

3. What is the general solution if an auxiliary equation has a repeated real root α?

   • (A + B) αⁿ
   • (A + Bn) αⁿ
   • A αⁿ + B n²
   • A αⁿ + B αⁿ

4. What form does the real solution take when auxiliary roots are complex numbers r e^{±iθ}?

   • rⁿ (A cos nθ + B sin nθ)
   • (A + Bn) rⁿ cos nθ
   • rⁿ (A cos θ + B sin θ)
   • A rⁿ cos nθ

5. What is the general solution to the recurrence relation u_{n+2} - 4u_{n+1} + 4u_n = 0?

   • (A + Bn) 4ⁿ
   • (A + Bn) 2ⁿ
   • A 4ⁿ + B 1ⁿ
   • A 2ⁿ + B (-2)ⁿ

6. Given the recurrence relation u_{n+1} = 3u_n + 2 with u_1 = 1, what is u_3?

   • 17
   • 53
   • 14
   • 5

7. Which condition defines a sequence u_n as periodic with period k?

   • u_{n+1} = k u_n
   • u_{n+k} = 0
   • u_{n+k} = u_n
   • u_{nk} = u_n

8. What is the period of the sequence defined by u_1 = 2 and u_{n+1} = 1/u_n?

   • 2
   • 1
   • 3
   • 4

9. For u_{n+1} - 2u_n = 5(3ⁿ), what trial particular solution should be used?

   • k 2ⁿ
   • k n 3ⁿ
   • k + 3ⁿ
   • k 3ⁿ

10. Which inequality defines a strictly increasing sequence u_n for all positive integers n?

    • u_{n+1} < u_n
    • u_{n+1} ≥ u_n
    • u_{n+1} = u_n + 1
    • u_{n+1} > u_n

11. What are the roots of the auxiliary equation for u_{n+2} - u_{n+1} - 6u_n = 0?

    • 1 and -6
    • -3 and 2
    • 3 and -2
    • 6 and -1

12. What is the general solution of the recurrence relation u_{n+2} - 7u_{n+1} + 12u_n = 0?

    • A 7ⁿ + B 12ⁿ
    • (A + Bn) 3ⁿ
    • A 3ⁿ + B 4ⁿ
    • A (-3)ⁿ + B (-4)ⁿ

13. For u_{n+2} - 5u_{n+1} + 6u_n = 2n + 1, what trial particular solution form is appropriate?

    • kn + c
    • kn² + c
    • k
    • k 2ⁿ + c

14. What condition on auxiliary roots α and β ensures u_n → 0 as n → ∞?

    • α β < 1
    • α < 1 and β < 1
    • α + β < 0
    • |α| < 1 and |β| < 1

15. What trial particular solution should be used for u_{n+1} - 2u_n = 2ⁿ?

    • k 2ⁿ
    • k n 2ⁿ
    • k 2ⁿ⁺¹
    • k n + 2ⁿ

16. What is the constant particular solution to the recurrence relation u_{n+1} - 3u_n = 8?

    • -2
    • 8
    • -4
    • 4

17. What is the least upper bound of the sequence u_n = n/(n+1) for n ≥ 1?

    • 0
    • Infinity
    • 1
    • 2

18. What is the general solution of the recurrence relation u_{n+2} + u_n = 0?

    • A cos(nπ/2) + B sin(nπ/2)
    • (A + Bn) iⁿ
    • A 1ⁿ + B (-1)ⁿ
    • A cos(nπ) + B sin(nπ)

19. What is the explicit formula for u_n given u_1 = 2 and u_{n+1} = u_n + 5?

    • 5ⁿ⁻¹ + 2
    • 5n + 2
    • 2n + 3
    • 5n - 3

20. If u_n = kn satisfies u_{n+1} - u_n = 4 for all n, what is k?

    • 2
    • 1
    • 8
    • 4

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