4D.6.1 Modelling with recurrence relations Quiz: Pearson Edexcel Further Maths, Unit 45

The 20 questions

1. How many previous terms directly determine u_(n+1) in a first-order recurrence relation?

   • 1
   • 0
   • n
   • 2

2. What is the purpose of an initial condition in a recurrence relation model?

   • Determines the order
   • Eliminates the variable
   • Finds steady states
   • Fixes arbitrary constants

3. A population grows by 5% a year and 200 are removed each year. Which recurrence models this, with P_n the population after n years?

   • P_(n+1) = 1.05 P_n - 200
   • P_(n+1) = 0.05 P_n - 200
   • P_(n+1) = P_n + 0.05 - 200
   • P_(n+1) = 1.05 P_n + 200

4. For u_(n+1) = a u_n + b, what equation does a steady state u satisfy?

   • u = b - a
   • u = a u + b
   • u = a + b
   • u = u + 1

5. What is the complementary function of a non-homogeneous recurrence relation?

   • Steady state value
   • Particular solution
   • Homogeneous general solution
   • Initial starting value

6. What is the auxiliary equation of u_(n+2) - 5u_(n+1) + 6u_n = 0?

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

7. Which mathematical feature defines a discrete-time recurrence relation model?

   • Infinite derivative
   • Smooth integration
   • Step-by-step change
   • Continuous rate

8. A population P_0 = 1000 follows P_(n+1) = 1.05 P_n - 200. What is P_1?

   • 950
   • 800
   • 850
   • 1050

9. For the same model, what is P_2?

   • 700
   • 722.5
   • 692.5
   • 850

10. For P_(n+1) = 1.05 P_n - 200, what is the steady state?

    • 1000
    • 5000
    • 200
    • 4000

11. A balance follows B_(n+1) = 1.02 B_n - 150 with B_0 = 2000. What is B_1?

    • 2040
    • 1890
    • 1950
    • 1850

12. For the population model P_(n+1) = 1.05 P_n - 200 with P_0 = 1000, what is P_3?

    • 527.125
    • 527.5
    • 600
    • 692.5

13. A savings account earns 4% a year and £100 is added at the end of each year. Which recurrence models the balance u_n?

    • u_(n+1) = 1.04 u_n + 100
    • u_(n+1) = 1.4 u_n + 100
    • u_(n+1) = 1.04 u_n - 100
    • u_(n+1) = u_n + 4 + 100

14. For the savings model u_(n+1) = 1.04 u_n + 100 with u_0 = 1000, what is u_2?

    • 1285.6
    • 1240
    • 1300
    • 1180

15. For the model u_(n+1) = 0.8 u_n + 50, what is the steady state?

    • 62.5
    • 250
    • 200
    • 50

16. For P_(n+1) = 1.05 P_n - 200 with P_0 = 1000, what is the value of P_1?

    • 950
    • 850
    • 800
    • 1050

17. Why is the steady state of the relation P_(n+1) = 1.05 P_n - 200 unstable?

    • Multiplier exceeds 1
    • Multiplier equals 1
    • Multiplier is negative
    • Constant is negative

18. To what value does the sequence u_(n+1) = 0.9 u_n + 30 converge?

    • 300
    • 30
    • 3000
    • 33

19. How many initial conditions are required to fix a unique solution for a second-order recurrence relation?

    • 0
    • 1
    • 3
    • 2

20. Why does u_(n+1) = u_n + 5 have no steady state?

    • No solution exists
    • Multiplier is zero
    • Initial value missing
    • Sequence is constant

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