Question

Demonstrati ca : $\frac{6}{9} \ \textless \ \frac{1}{31} + \frac{1}{32} +...+ \frac{1}{90} \ \textless \ 2$

Answer 1

1/31 <1/30
1/32  <1/30
1/33  <1/30
   .
   .
   .
1/90< 1/30
Daca le adunam:
1/31+1/32+1/33+....+1/90 <1/30+1/30+1/30+...+1/30(de 60 de ori)
1/31+1/32+1/33+..+1/90< 60*1/30
1/31+1/32+1/33+...+1/90<2 (1)

Facem la fel:
1/31 >1/90
1/32 >1/90
1/33> 1/90
    .
    .
    .
1/90≥1/90
Daca le adunam:
1/31+1/32+1/33+...+1/90 >1/90+1/90+1/90+...+1/90 (de 60 de ori)
1/31+1/32+1/33+...+1/90  > 60*1/90
 1/31+1/32+1/33+...+1/90  >6/9 (2)
Din (1) si (2) rezulta concluzia problemei.