4D.6.2 Solving first and second order recurrence relations Quiz: Pearson Edexcel Further Maths, Unit 45

The 20 questions

1. What is the auxiliary equation of a first-order recurrence u_(n+1) = a u_n?

   • m^2 = a
   • m = 1/a
   • m = a
   • m = a + 1

2. For u_(n+1) - a u_n = b with a not equal to 1, what constant particular solution c is used?

   • c = a b
   • c = b / (a - 1)
   • c = b / (1 - a)
   • c = b(1 - a)

3. What is the general solution of the homogeneous first-order recurrence u_(n+1) = a u_n?

   • u_n = A n^a
   • u_n = A a^n
   • u_n = A + a n
   • u_n = A a + n

4. For a second-order homogeneous recurrence with two distinct real roots m1 and m2, what is the general solution?

   • (A + Bn) m1^n
   • A m1^n - B m2^n with A = B
   • A m1 + B m2
   • A m1^n + B m2^n

5. For a second-order homogeneous recurrence with a repeated root m, what is the general solution?

   • (A + Bn) m^n
   • A + B m^n
   • A m^n + B n
   • A m^n + B m^n

6. What do the complementary function and the particular solution describe?

   • Two different auxiliary equations.
   • The homogeneous part of the general solution, and a specific solution of the full recurrence.
   • The solution with n = 0 only.
   • The initial values and the final values only.

7. How many arbitrary constants does the general solution of a second-order recurrence contain?

   • 1
   • 3
   • 0
   • 2

8. For u_(n+1) - 5u_n = 8 with u_1 = 1, what is u_4?

   • 423
   • 375
   • 73
   • 373

9. For u_(n+1) - 5u_n = 8 with u_1 = 1, which formula gives the general term?

   • u_n = 3 × 5^n - 2
   • u_n = 3 × 5^(n-1) + 8
   • u_n = 3 × 5^(n-1) - 2
   • u_n = 5^n - 2

10. What are the roots of the auxiliary equation for u_(n+2) - 3u_(n+1) + 2u_n = 0?

    • m = 3 and m = 2
    • m = -1 and m = -2
    • m = 1 and m = 2
    • m = 1 only, as a repeated root

11. What is the general solution of u_(n+2) - 3u_(n+1) + 2u_n = 0?

    • u_n = A + B × 2^n
    • u_n = (A + Bn) × 2^n
    • u_n = A + B × 3^n
    • u_n = A × 3^n + B × 2^n

12. For u_(n+2) - 4u_(n+1) + 4u_n = 0, what is the general solution?

    • u_n = A × 4^n + B × 2^n
    • u_n = (A + Bn) × 4^n
    • u_n = (A + Bn) × 2^n
    • u_n = A × 2^n + B × 2^n

13. For u_(n+2) - 5u_(n+1) + 6u_n = 12, what is the general solution, using a constant particular solution?

    • u_n = (A + Bn) × 2^n + 6
    • u_n = A × 2^n + B × 3^n
    • u_n = A × 2^n + B × 3^n + 6
    • u_n = A × 2^n + B × 3^n + 12

14. For u_(n+1) = 0.5 u_n + 3 with u_0 = 10, what is u_2?

    • 8
    • 6
    • 7
    • 9

15. For u_(n+1) = 0.5 u_n + 3, what value does u_n approach in the long run?

    • 10
    • 0
    • 3
    • 6

16. For u_(n+2) = 3u_(n+1) - 2u_n with u_0 = 0 and u_1 = 1, which formula gives u_n?

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

17. Why is the constant particular solution of u_(n+1) = 5u_n + 8 equal to -2?

    • Zero satisfies any recurrence with a positive constant.
    • The particular solution is always zero when b is positive.
    • It must satisfy c = 5c + 8, which gives c = -2.
    • Because u_1 = 1 forces c = 1.

18. For u_(n+2) - 5u_(n+1) + 6u_n = 0 with u_0 = 1 and u_1 = 5, what is u_2?

    • 7
    • 19
    • 25
    • 13

19. What happens to u_n for u_(n+1) = 5u_n + 8 with u_1 = 1 as n grows?

    • It converges to -2, the particular solution.
    • It converges to 8 divided by 5.
    • It grows without bound, since the factor 5 exceeds 1.
    • It oscillates between 1 and -2.

20. Which equation is a first-order non-homogeneous recurrence?

    • u_(n+1) = u_n^2 + 1
    • u_(n+2) = 3u_n
    • u_(n+2) = 2u_(n+1) + u_n
    • u_(n+1) = 2u_n + 5

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