Question

Sa se arate ca numarul N=0,5+0,5^2+0.5^3+.....+0,5^50+2×0,5^51 este numar natural.

Answer 1

(1/2)(1/2+...(1/2)^50)+(1/2)^51=(1/2) *( 1-(1/2)^51)/(1-1/2) +(1.2)^51=
1-(1/2)^51+(1/2)^51=1∈N

Answer 2

S=0,5+0,5^2+0,5^3+.........+0,5^50 | ×0,5
0,5×S=0,5^2+0,5^3+...........+0,5^51
scăzând cele două sume
0,5S-S=0,5^2-0,5-0,5^2+0,5^3-0,5^3+.........+0,5^51
-0,5S=0,5^51-0,5 | (-0,5)
S=-0,5^50+1
N=-0,5^50+1+2×0,5×0,5^50
N=-0,5^50+1+2×1/2×0,5^50
N=-0,5^50+1+0,5^50
=1 Nr natural