Question

Arătați ca: [x] + [x+1/2]=[2x]. Etapele si explicatiile rezolvării,va rog!

Answer 1

$\displaystyle Notam~[x]=k \in \mathbb{Z}~si~\{x\}=r \in [0,1).~Atunci~x=k+r. \\ \\ Cum~r \in [0,1),~vom~avea~doua~cazuri: \\ \\ \bullet~Cazul~1:~r \in \left [ 0, \frac{1}{2} \right).~In~acest~caz~avem~x+ \frac{1}{2} \in \left [k+ \frac{1}{2}, k+1 \right), \\ \\ deci~ \left[ x+ \frac{1}{2 }\right]=k. \\ \\ Totodata~2x=2k+2r \in [2k,2k+1),~deci~[2x]=2k. \\ \\ Deci~avem~[x]+ \left[x+ \frac{1}{2} \right]=2k=[2x].$

$\displaystyle \bullet ~Cazul~2:~r \in \left[ \frac{1}{2},1 \right).~In~acest~caz~avem~x+ \frac{1}{2} \in \left[k+1,k+ \frac{3}{2} \right), \\ \\ deci~ \left[ x+ \frac{1}{2} \right]=k+1. \\ \\ Totodata~2x=2k+2r \in [2k+1,2k+2),~deci~[2x]=2k+1. \\ \\ Deci~avem~[x]+ \left[x+\frac{1}{2} \right]=2k+1=[2x].$