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Question Number 159122 by physicstutes last updated on 13/Nov/21

In order to monitor buses in a travel agency, the manager decides to monitor the number of break downs of the buses using the sequence {x_n } defined by x_(n+1) = 1.05 x_n + 4. Given that x_0 = 40. is the number of break downs by the buses from the 1^(st) of january 2000, and that for every n∈N, we denote x_n the number of breakdowns of the buses as from 1^(st) of january of the year (2000 + n) (a) Calculate x_1 , x_2 , x_3 (b) Consider the sequence {y_n } defined by y_n = x_n + 80 for all n ∈ N (i) express y_(n+1) in terms of y_n and deduce the nature of the sequence {y_n }. (ii) Express y_n in terms of n. deduce x_n in terms of n (iv) find the number of break downs that will be registered by 1^(st) january 2021.

$$\mathrm{In}\:\mathrm{order}\:\mathrm{to}\:\mathrm{monitor}\:\mathrm{buses}\:\mathrm{in}\:\mathrm{a}\:\mathrm{travel} \\ $$$$\mathrm{agency},\:\mathrm{the}\:\mathrm{manager}\:\mathrm{decides}\:\mathrm{to}\:\mathrm{monitor} \\ $$$$\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{break}\:\mathrm{downs}\:\mathrm{of}\:\mathrm{the}\:\mathrm{buses} \\ $$$$\mathrm{using}\:\mathrm{the}\:\mathrm{sequence}\:\left\{{x}_{{n}} \right\}\:\mathrm{defined}\:\mathrm{by} \\ $$$${x}_{{n}+\mathrm{1}} \:=\:\mathrm{1}.\mathrm{05}\:{x}_{{n}} \:+\:\mathrm{4}.\:\mathrm{Given}\:\mathrm{that}\:{x}_{\mathrm{0}} \:=\:\mathrm{40}. \\ $$$$\mathrm{is}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{break}\:\mathrm{downs}\:\mathrm{by}\:\mathrm{the}\:\mathrm{buses} \\ $$$$\mathrm{from}\:\mathrm{the}\:\mathrm{1}^{\mathrm{st}} \:\mathrm{of}\:\mathrm{january}\:\mathrm{2000},\:\mathrm{and}\:\mathrm{that} \\ $$$$\mathrm{for}\:\mathrm{every}\:{n}\in\mathbb{N},\:\mathrm{we}\:\mathrm{denote}\:{x}_{{n}} \:\mathrm{the}\:\mathrm{number} \\ $$$$\mathrm{of}\:\mathrm{breakdowns}\:\mathrm{of}\:\mathrm{the}\:\mathrm{buses}\:\mathrm{as}\:\mathrm{from}\:\mathrm{1}^{\mathrm{st}} \\ $$$$\mathrm{of}\:\mathrm{january}\:\mathrm{of}\:\mathrm{the}\:\mathrm{year}\:\left(\mathrm{2000}\:+\:{n}\right) \\ $$$$\left(\mathrm{a}\right)\:\mathrm{Calculate}\:{x}_{\mathrm{1}} ,\:{x}_{\mathrm{2}} \:,\:{x}_{\mathrm{3}} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{Consider}\:\mathrm{the}\:\mathrm{sequence}\:\left\{{y}_{{n}} \right\}\:\mathrm{defined} \\ $$$$\mathrm{by}\:{y}_{{n}} \:=\:{x}_{{n}} \:+\:\mathrm{80}\:\mathrm{for}\:\mathrm{all}\:{n}\:\in\:\mathbb{N} \\ $$$$\left(\mathrm{i}\right)\:\mathrm{express}\:{y}_{{n}+\mathrm{1}} \:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:{y}_{{n}} \:\mathrm{and} \\ $$$$\mathrm{deduce}\:\mathrm{the}\:\mathrm{nature}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sequence}\:\left\{{y}_{{n}} \right\}. \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{Express}\:{y}_{{n}} \:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:{n}.\:\mathrm{deduce}\:{x}_{{n}} \\ $$$$\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:{n} \\ $$$$\left(\mathrm{iv}\right)\:\mathrm{find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{break}\:\mathrm{downs} \\ $$$$\mathrm{that}\:\mathrm{will}\:\mathrm{be}\:\mathrm{registered}\:\mathrm{by}\:\mathrm{1}^{\mathrm{st}} \:\mathrm{january}\: \\ $$$$\mathrm{2021}. \\ $$

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