Question Number 75569 by naka3546 last updated on 12/Dec/19
Commented by MJS last updated on 13/Dec/19
$$\mathrm{was}\:\mathrm{my}\:\mathrm{answer}\:\mathrm{to}\:\mathrm{the}\:\mathrm{other}\:\mathrm{one}\:\mathrm{ok}? \\ $$$$\mathrm{it}'\mathrm{s}\:\mathrm{hard}\:\mathrm{to}\:\mathrm{understand}... \\ $$
Commented by naka3546 last updated on 13/Dec/19
$$\mathrm{2019}^{{n}} \:\:=\:\:.....\:{abcde} \\ $$$$\mathrm{2019}^{{m}} \:\:=\:\:.....\:{abcde} \\ $$$${min}\:\left\{{m}+{n}\right\}\:=\:? \\ $$
Commented by MJS last updated on 13/Dec/19
$$\mathrm{so}\:\mathrm{we}\:\mathrm{have} \\ $$$${a}_{{n}+\mathrm{1}} =\mathrm{fpart}\:\left(\mathrm{2019}{a}_{{n}} \right) \\ $$$${a}_{\mathrm{0}} =.\mathrm{00001} \\ $$$$\mathrm{and}\:\mathrm{we}'\mathrm{re}\:\mathrm{asking}\:\mathrm{for}\:\mathrm{the}\:\mathrm{length}\:\mathrm{of}\:\mathrm{the}\:\mathrm{period} \\ $$$$\mathrm{of}\:\mathrm{this}\:\mathrm{series} \\ $$
Commented by MJS last updated on 13/Dec/19
$$\mathrm{I}'\mathrm{m}\:\mathrm{testing} \\ $$$${a}_{\mathrm{0}} =.\mathrm{00001} \\ $$$${a}_{\mathrm{50}} =.\mathrm{89001} \\ $$$${a}_{\mathrm{100}} =.\mathrm{78001} \\ $$$$... \\ $$$${a}_{\mathrm{500}} =.\mathrm{90001} \\ $$$$\Rightarrow\:{a}_{\mathrm{5000}} =.\mathrm{00001} \\ $$$$\mathrm{I}\:\mathrm{think}\:\mathrm{this}\:\mathrm{goes}\:\mathrm{for}\:\mathrm{any}\:{a}_{{n}} =.{bcdef} \\ $$$${a}_{{n}} ={a}_{{n}+\mathrm{5000}} =.{bcdef} \\ $$$$\Rightarrow\:\mathrm{min}\:\left({m}+{n}\right)\:=\mathrm{5000}\:\mathrm{if}\:\mathrm{we}\:\mathrm{allow}\:{m}=\mathrm{0} \\ $$$$\mathrm{or}\:\mathrm{min}\:\left({m}+{n}\right)\:=\mathrm{5002}\:\mathrm{if}\:{m}>\mathrm{0} \\ $$