Texts: Exercise 7
As of 1 January 2022, the board of directors of the telephone operator named A studied the evolution of the number of subscribers in a given country with respect to its main competitor, telephone operator named B. In January 2022, each of A and B had 300,000 subscribers.
For all n ∈ N, we denote by a the number of subscribers, in thousands, for operator A, and b represents the number of subscribers, in thousands, for operator B in the year 2022+n.
Let a₀ = 300 and b₀ = 300. An analysis of the statistical data collected over several years led to the following information:
aₙ₊₁ = 0.7aₙ + 0.2bₙ + 60
bₙ₊₁ = 0.1aₙ + 0.6bₙ + 70 (1)
Let U = [a, b]ᵀ. Using Eq. (1), determine Uᵢ. Find the matrices M and P such that Eq. (1) can be expressed in matrix form as follows:
U = MU + P
Given that Q = [a, b]ᵀ, calculate I - MQ. Find I - M⁻¹. Hence, or otherwise, find U such that
U = MU + P
For all n ∈ N, let V = U - Uₙ. Deduce that
Vₙ₊₁ = MVₙ
Hence, or otherwise, show that
V = MV₀
Given that for all n ∈ N,
100 0.8³ 50 0.8³
140 0.5³ 140 0.5³
V = [a, b]ᵀ
Express U in terms of n.
Determine lim n and lim b. Give an interpretation of the result.
