Recurrence relations

Determine, in terms of q, the general solution of the first-order recurrence relation.

Hence explain why, when q is close to zero, G_n is approximately constant and equal to

1.25.

In the case where G₁ = 1.22 and G₂ = 1.2378, find the values of q and G₀.

The population of a species of fish F_n is being modelled by the second-order recurrence relation

F_n+2 = 1.25F_n+1 - rF_n+1F_n,

where r is a positive constant and n is the number of weeks since the start of the breeding season.

You are given that the value of r is such that the model predicts that the population approaches a limiting value, L.

Determine L in terms of r.

Determine what happens if a term of the sequence is greater than

Explain why the modified recurrence relation F_n+2 = INT(1.25F_n+1 - rF_n+1F_n) may be

preferable when modelling this fish population.

When r = 0.0001, both F_n+2 = 1.25F_n+1 - rF_n+1F_n and F_n+2 = INT(1.25F_n+1 - rF_n+1F_n) predict that the fish population approaches a limiting value.

By finding the limiting value for each model, when F₀ = 2450 and F₁ = 2475, compare

the two models for the fish population.

Write down the values of α + β and αβ.

Consider the sequence {S_n}, where S_n = αⁿ + βⁿ for n ⩾ 0.

Determine the values of S₂ and S₃.

Show that S_n+2 = S_n+1 + S_n for n ⩾ 0.

Deduce that S_n is an integer for all n ⩾ 0.

A student models the terms of the sequence {S_n} using the formula T_n = αⁿ.

Explain why this formula is unsuitable for every n ⩾ 1.

Considering the cases n even and n odd separately, state a modification of the formula

T_n = αⁿ, other than T_n = αⁿ + βⁿ, such that T_n = S_n for all n ⩾ 1.

The irrational number φ = 1 + plays a significant role in the sequence of Fibonacci

Prove by induction that, for each positive integer n, φⁿ = F_n × φ + F_n-1.

In a national park, the number of adults of a given species is carefully monitored and controlled. The number of adults, n months after the start of this project, is A_n. Initially, there are 1000 adults. It is predicted that this number will have declined to 960 after one month.

The first model for the number of adults is that, from one month to the next, a fixed proportion of adults is lost. In order to maintain a fixed number of adults, the park managers “top up” the numbers by adding a constant number of adults from other parks at the end of each month.

Use this model to express the number of adults as a first-order recurrence system.

Instead, it is found that, the proportion of adults lost each month is double the predicted amount, with no change being made to the constant number of adults added each month.

Show that the revised recurrence system for A_n is A₀ = 1000, A_n+1 = 0.92A_n + 40.

Solve this revised recurrence system.

Describe the long-term behaviour of the sequence {A_n} in this case.

A more refined model for the number of adults uses the second-order recurrence system A_n+1 = 0.9A_n - 0.1A_n-1 + 50, for n ⩾ 1, with A₀ = 1000 and A₁ = 920.

Determine the long-term behaviour of the sequence {A_n} for this more refined model.

A criticism of this more refined model is that it does not take account of the fact that the number of adults must be an integer at all times.

State a modified form of the second-order recurrence relation for this more refined

model that will satisfy this requirement.

{u_n} is constant.

{u_n} is periodic, with period 2.

{u_n} is periodic, with period 4.