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Question-75569




Question Number 75569 by naka3546 last updated on 12/Dec/19
Commented by MJS last updated on 13/Dec/19
was my answer to the other one ok?  it′s hard to understand...
$$\mathrm{was}\:\mathrm{my}\:\mathrm{answer}\:\mathrm{to}\:\mathrm{the}\:\mathrm{other}\:\mathrm{one}\:\mathrm{ok}? \\ $$$$\mathrm{it}'\mathrm{s}\:\mathrm{hard}\:\mathrm{to}\:\mathrm{understand}… \\ $$
Commented by naka3546 last updated on 13/Dec/19
2019^n   =  ..... abcde  2019^m   =  ..... abcde  min {m+n} = ?
$$\mathrm{2019}^{{n}} \:\:=\:\:…..\:{abcde} \\ $$$$\mathrm{2019}^{{m}} \:\:=\:\:…..\:{abcde} \\ $$$${min}\:\left\{{m}+{n}\right\}\:=\:? \\ $$
Commented by MJS last updated on 13/Dec/19
so we have  a_(n+1) =fpart (2019a_n )  a_0 =.00001  and we′re asking for the length of the period  of this series
$$\mathrm{so}\:\mathrm{we}\:\mathrm{have} \\ $$$${a}_{{n}+\mathrm{1}} =\mathrm{fpart}\:\left(\mathrm{2019}{a}_{{n}} \right) \\ $$$${a}_{\mathrm{0}} =.\mathrm{00001} \\ $$$$\mathrm{and}\:\mathrm{we}'\mathrm{re}\:\mathrm{asking}\:\mathrm{for}\:\mathrm{the}\:\mathrm{length}\:\mathrm{of}\:\mathrm{the}\:\mathrm{period} \\ $$$$\mathrm{of}\:\mathrm{this}\:\mathrm{series} \\ $$
Commented by MJS last updated on 13/Dec/19
I′m testing  a_0 =.00001  a_(50) =.89001  a_(100) =.78001  ...  a_(500) =.90001  ⇒ a_(5000) =.00001  I think this goes for any a_n =.bcdef  a_n =a_(n+5000) =.bcdef  ⇒ min (m+n) =5000 if we allow m=0  or min (m+n) =5002 if m>0
$$\mathrm{I}'\mathrm{m}\:\mathrm{testing} \\ $$$${a}_{\mathrm{0}} =.\mathrm{00001} \\ $$$${a}_{\mathrm{50}} =.\mathrm{89001} \\ $$$${a}_{\mathrm{100}} =.\mathrm{78001} \\ $$$$… \\ $$$${a}_{\mathrm{500}} =.\mathrm{90001} \\ $$$$\Rightarrow\:{a}_{\mathrm{5000}} =.\mathrm{00001} \\ $$$$\mathrm{I}\:\mathrm{think}\:\mathrm{this}\:\mathrm{goes}\:\mathrm{for}\:\mathrm{any}\:{a}_{{n}} =.{bcdef} \\ $$$${a}_{{n}} ={a}_{{n}+\mathrm{5000}} =.{bcdef} \\ $$$$\Rightarrow\:\mathrm{min}\:\left({m}+{n}\right)\:=\mathrm{5000}\:\mathrm{if}\:\mathrm{we}\:\mathrm{allow}\:{m}=\mathrm{0} \\ $$$$\mathrm{or}\:\mathrm{min}\:\left({m}+{n}\right)\:=\mathrm{5002}\:\mathrm{if}\:{m}>\mathrm{0} \\ $$

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