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

The 20 questions

1. What does a first-order recurrence relation u_(n+1) = a u_n + b model?

   • A process in which each term is determined from the previous term.
   • A random sequence with no rule.
   • Only geometric series with a = 1.
   • A set of simultaneous equations solved all at once.

2. What is the initial condition in a recurrence model?

   • The limit of the terms as n tends to infinity.
   • The sum of all the terms so far.
   • The value of the constant b only.
   • The given starting value, such as u_0 or u_1, needed to fix the solution.

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?

   • The sum of the initial conditions.
   • The particular solution found by trying a constant.
   • The general solution of the homogeneous recurrence with the constant term set to zero.
   • The solution of the auxiliary equation with b added.

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. Why is modelling with recurrence relations useful?

   • It replaces all data collection.
   • It works only for constant growth.
   • It describes how a quantity changes from one period to the next, such as yearly population.
   • It gives the exact value at any time without starting values.

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 the population model P_(n+1) = 1.05 P_n - 200 starting at 1000, which way does the population move in the first year?

    • It rises and then returns to 1000.
    • It falls, since P_1 = 850 is below 1000.
    • It stays at 1000.
    • It rises, since 5% growth outweighs the removal.

17. Why is the steady state 4000 of P_(n+1) = 1.05 P_n - 200 unstable?

    • Above 4000 the population keeps rising and below 4000 it keeps falling, so the sequence moves away from 4000.
    • It is stable because 1.05 is less than 1.
    • It is the first term of the sequence.
    • Because b is negative, it attracts all sequences.

18. Does the sequence u_(n+1) = 0.9 u_n + 30 converge from any starting value?

    • Yes, it converges to 300 from any starting value, since the multiplier 0.9 has absolute value below 1.
    • No, it diverges from any starting value above 300.
    • No, it oscillates between two values for every start.
    • Only if the starting value is exactly 300.

19. Why must a recurrence model include an initial condition?

    • It removes the need for a steady state.
    • It is needed only when a is negative.
    • It makes b equal to zero.
    • Without one, infinitely many sequences satisfy the recurrence, so the actual quantity cannot be determined.

20. Why does a steady state fail to exist when a = 1 and b is not zero?

    • The equation u = u + b has no solution, so there is no fixed value for the sequence to settle at.
    • Because b must be zero when a is 1, so the steady state is always zero.
    • Because the population must double each year.
    • Because a = 1 always gives an oscillating sequence.

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