Răspuns:
S1=9/10
S2=47/260
Explicație pas cu pas:
S1=1/2+1/6+1/12+1/20+.....+1/90=
1/(1×2)+1/(2×3)+1/(3×4)+1/(4×5)+....+1/(9×10)
Acum:
1/(1×2)=1/1-1/2
1/(2×3)=1/2-1/3
1/(3×4)=1/3-1/4
...
1/(9×10)=1/9-1/10
Inlocuind, obtinem:
S1=(1/1-1/2)+(1/2-1/3)+(1/3-1/4)+...+(1/9-1/10)=
Termenii -1/2 si 1/2, .... se reduc.
Deci
S1=1-1/10=9/10
---------
Similar procedam pentru S2:
1/30=1/5-1/6
1/42=1/6-1/7
1/56=1/7-1/8
....
1/2652=1/51-1/52
Dupa reducerea termenilor, ca la S1, avem
Deci S2=1/5-1/52=(52-5)/(5×52)=47/160
