Question

Aratati ca x este numar rational:

Answer 1

Răspuns:

folosim descompunerea:

1/a-1/(a+1)=(a+1-a)/a(a+1)=1/a(a+1)

Explicație pas cu pas:

X=√1/94[(1/1×2+1/2×3+...+1/24×25)-(1/25×26+1/26×27+..+1/49×50)]=

√1/94[(1/1-1/2+1/2-1/3+1/3-1/4+.. +1/24-1/25)-

(1/25-1/26+1/26-1/27+...+1/49-1/50)]=

√1/94[(1-1/25)-(1/25-1/50)]=

√1/94(1-1/25-1/25+1/50)=

√1/94(⁵⁰1-²2/25+1/50)=

√1/94(50-4+1)=√[1/(2×47)]×(47/50)=

√1/100=1/10 aparține mulțimii Q