Question Number 196070 by sniper237 last updated on 17/Aug/23
$${During}\:{an}\:{invasion}\: \\ $$$${In}\:{the}\:{amazon}\:{dominion} \\ $$$${Every}\:{evening}\:,\:{every}\:\:{invader}\: \\ $$$${kills}\:\:{an}\:\:{amazon}\:{warrior}\: \\ $$$${Every}\:{morning}\:,\:{every}\:{amazon} \\ $$$${kills}\:\:\:{an}\:{invader}\:{soldier} \\ $$$${The}\:\mathrm{8}^{{th}} \:{day}\:{evening}\:,\:{there}\:{remains}\: \\ $$$${only}\:{one}\:{amazon}\:\:{and}\:{no}\:{invader} \\ $$$$ \\ $$$${How}\:{many}\:\:{were}\:{they}\:{in}\:{each}\:{part}\:? \\ $$
Answered by jabarsing last updated on 18/Aug/23
$$\mathrm{1597}\:{amazon} \\ $$$$\mathrm{987}\:{invader}\: \\ $$
Commented by sniper237 last updated on 18/Aug/23
$${That}'{s}\:{right}\: \\ $$$$\begin{cases}{{x}_{{n}+\mathrm{1}} \:=\:{x}_{{n}} −{y}_{{n}} }\\{{y}_{{n}+\mathrm{1}} =\:{y}_{{n}} −{x}_{{n}+\mathrm{1}} }\end{cases}{and}\:\:{x}_{\mathrm{8}} =\mathrm{1}\:,\:{y}_{\mathrm{8}} =\mathrm{0} \\ $$$${V}_{{n}+\mathrm{1}} \:=\:{AV}_{{n}} \:\:\:\:\:\Rightarrow\:{V}_{\mathrm{8}} \:=\:{A}^{\mathrm{8}} {V}_{\mathrm{0}} \: \\ $$$$\begin{pmatrix}{{x}_{\mathrm{0}} }\\{{y}_{\mathrm{0}} }\end{pmatrix}=\begin{pmatrix}{\mathrm{1}\:\:\:\:\:−\mathrm{1}}\\{−\mathrm{1}\:\:\:\:\:\mathrm{2}}\end{pmatrix}^{−\mathrm{8}} \begin{pmatrix}{{x}_{\mathrm{8}} }\\{{y}_{\mathrm{8}} }\end{pmatrix}\: \\ $$