Question

Cho tổng S= \(\dfrac{1}{31}\) + \(\dfrac{1}{32}\) + ... + \(\dfrac{1}{60}\) Chứng minh \(\dfrac{3}{5}\) < S < \(\dfrac{4}{5}\)

Answer 1

1/31>1/40

1/32>1/40

...

1/40=1/40

=>1/31+1/32+...+1/40>1/40*10=1/4

1/41>1/50

1/42>1/50

...

1/50=1/50

=>1/41+1/42+...+1/50>10/50=1/5

1/51>1/60

1/52>1/60

...

1/60=1/60

=>1/51+1/52+...+1/60>10/60=1/6

=>S>1/4+1/5+1/6=3/5

1/31<1/30

1/32<1/30

...

1/40<1/30

=>1/31+1/32+...+1/40<1/30*10=1/3

1/41<1/40

1/42<1/40

...

1/50<1/40

=>1/41+1/42+...+1/50<10/40=1/4

1/51<1/50

1/52<1/50

...

1/60<1/50

=>1/51+1/52+...+1/60<10/50=1/5

=>S<1/3+1/4+1/5=4/5