Question

Va rog frumos daca ma puteti ajuta!!!!!Multumesc.

Fie expresia E(x)=|x-100|+|x-99|+...+|x-1|+|x+1|+...+ |x+99|+|x+100|.

a) Aratati ca E(101) = E(-101) si calculati aceasta valoare.
b) Stabiliti daca E(0) este numar par;
c) Aratati ca daca a ≥ 102, atunci E(a)=E(-a)

Answer 1

E(101)=1+2+...+201=201*202/2=201*101=20301
E(-101)=201+200+...+1=E(101)

E(0)= |-100|+...+|-1|+|1|+...+|100|= 2*(|-100|+...+|100|), divizibil cu 2, deci par


c)pt a≥102
E(a)=a-100+a-99+...+a-1+a+1+a+2+...a+100=200a
E(-a)=|-a-100|+|-a-99|+...+|-a-1|+|-a+1|+|-a+2|+...+|-a+100|=
          a+100+a+99+...+a+1 +a-1+a-2+.....+a-100=200a=E(a) , C.C.T.D.