Question

aratati ca primele 2013 zecimale ale nr.√0,444...4(cifra 4 apare de 2013 ori)sunt egale cu 6
indicatie:aratam ca 0,666...6<√0,444...4<0(6)

Answer 1

$Sa~notam~numarul~cu~n. \\ \\ n\ \textless \ \sqrt{0,(4)}=\sqrt{ \frac{4}{9}} = \frac{2}{3}=0,(6). \\ \\ Ramane~sa~demonstram~ca~n\ \textgreater \ 0,666...6~(6~se~repeta~de~2013~ori). \\ \\ Notam~ \underbrace{111...1} =a. \\ ~~~~~~~~~~2013~cifre \\ \\ n=\sqrt{ \frac{4a}{10^{2013}} }.$

$n\ \textgreater \ 0,666...6 \Leftrightarrow \sqrt{ \frac{4a}{10^{2013}} } \ \textgreater \ \frac{6a}{10^{2013}} \Leftrightarrow \frac{4a}{10^{2013}}\ \textgreater \ \frac{36a^2}{10^{2026}} \Leftrightarrow 1\ \textgreater \ \frac{9a}{10^{2013}} \Leftrightarrow \\ \\ \Leftrightarrow 10^{2013}\ \textgreater \ 9a \Leftrightarrow 1 \underbrace{000....0}\ \textgreater \ \underbrace{999...9},~evident. \\ ~~~~~~~~~~~~~~~~~~~~~~2013~cifre~~2013~cifre$

$Astfel~am~domentrat~ca~0, \underbrace{666...6}\ \textless \ n\ \textless \ 0,(6),~de~unde~rezulta~ca \\~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~2013~cifre \\ primele~2013~zecimale~ale~lui~n~sunt~egale~cu~6.$