Question

[1/2] +[2/2] +....+[n/2]=? Va rog ajutați-mă unde [] e parte întregea

Answer 1

Observi  pentru  inceput  ca  [1/2]=0 Vom  face  abstractie  de  acest  termen.
Aduam  separat  termenii  pari(n=2k)  si  separat  cei  inpari (n=2k+1)
fie n=2k  ,k∈N*.k=[n/2]=n/2  relatia  (1.
S(2k) =[2/2]+[4/2]+[6/2]+...+[2k/2]=1+2+...+k=suma  Gauss=k*(k+1)/2 
n=2k+1
S(2k+1)=[3/2]+[5/2]+....[(2k+1)/2]=[2/2+1/2]+[4/2+1/2)]+...[2k/2+1/2]=
1+2+...+k=k(K+1)2
S=S(2k)+S(2k+1)=2*k*(k+1)/2=  k*(k+1)
Se  obseva  ca  [n/2]=[2n+1)/2]=[n/2]
Asadar
s=[n/2]*([n/2]+1)