Question

17.Calculati S¹=1 supra 2 plus 1 supra 6 plus 1 supra 12 plus 1 supra 20 +...+ 1 supra 90 S²=1 supra 30 plus 1 supra 42 plus 1 supra 56 plus 1 supra 72 +...+1 supra 2652​

Answer 1

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