S=1/30+1/31+1/32+1/33+...+1/59+1/60

S có 31 phân số,ta thấy:

1/30>1/62                             1/31>1/62                          1/32>1/62         ............          1/60>1/62

Vậy:

S>31.1/62

S>31/62

S>1/2

Vậy S>1/2

Chúc em học tốt^^

A>4/5

\(M=\dfrac{1}{31}+\dfrac{1}{32}+.................+\dfrac{1}{89}+\dfrac{1}{90}\)

\(\Leftrightarrow M=\left(\dfrac{1}{31}+\dfrac{1}{32}+.........+\dfrac{1}{60}\right)+\left(\dfrac{1}{61}+\dfrac{1}{62}+........+\dfrac{1}{90}\right)\)

Đặt :

\(A=\dfrac{1}{31}+\dfrac{1}{32}+.......+\dfrac{1}{60}>\dfrac{1}{60}+\dfrac{1}{60}+.......+\dfrac{1}{60}=\dfrac{1}{60}.30=\dfrac{1}{2}\)

\(B=\dfrac{1}{61}+\dfrac{1}{62}+.......+\dfrac{1}{90}>\dfrac{1}{90}+\dfrac{1}{90}+......+\dfrac{1}{90}=\dfrac{1}{90}.30=\dfrac{1}{3}\)

\(\Leftrightarrow M=\dfrac{1}{31}+\dfrac{1}{32}+.........+\dfrac{1}{89}+\dfrac{1}{90}=A+B< \dfrac{1}{2}+\dfrac{1}{3}=\dfrac{5}{6}\)

\(\Leftrightarrow M< \dfrac{5}{6}\rightarrowđpcm\)

(1/51 +1/52+  1/53 +1/54+ 1/55 +1/56+  1/57 +1/58 + 1/59 +1/60)(1/51 +1/52+  1/53 +1/54+ 1/55 +1/56+  1/57 +1/58 + 1/59 +1/60)(1/51 +1/52+  1/53 +1/54+ 1/55 +1/56+  1/57 +1/58 + 1/59 +1/60)(1/51 +1/52+  1/53 +1/54+ 1/55 +1/56+  1/57 +1/58 + 1/59 +1/60)(1/51 +1/52+  1/53 +1/54+ 1/55 +1/56+  1/57 +1/58 + 1/59 +1/60)(1/51 +1/52+  1/53 +1/54+ 1/55 +1/56+  1/57 +1/58 + 1/59 +1/60)(1/51 +1/52+  1/53 +1/54+ 1/55 +1/56+  1/57 +1/58 + 1/59 +1/60)(1/51 +1/52+  1/53 +1/54+ 1/55 +1/56+  1/57 +1/58 + 1/59 +1/60)(1/51 +1/52+  1/53 +1/54+ 1/55 +1/56+  1/57 +1/58 + 1/59 +1/60)(1/51 +1/52+  1/53 +1/54+ 1/55 +1/56+  1/57 +1/58 + 1/59 +1/60)(1/51 +1/52+  1/53 +1/54+ 1/55 +1/56+  1/57 +1/58 + 1/59 +1/60)

E = 1/31+1/32+...+1/60

E > 1/40+1/40+...+1/40+1/41+1/42+...+1/60

E > 20/40+1/41+1/42+...+1/60

E > 1/2+1/60+1/60+...+1/60

E > 1/2 + 1/3 = 5/6

Mà 5/6 > 4/5

=> E > 4/5

E = (1/31 +1/32+  1/33 +1/34+ 1/35 +1/36+  1/37 +1/38 + 1/39 +1/40) +

( 1/41 +1/42+  1/43 +1/44+ 1/45 +1/46+  1/47 +1/48 + 1/49 +1/50)+

(1/51 +1/52+  1/53 +1/54+ 1/55 +1/56+  1/57 +1/58 + 1/59 +1/60)

Mà  (1/31 +1/32+  1/33 +1/34+ 1/35 +1/36+  1/37 +1/38 + 1/39 +1/40) < ( 1/31 . 10) = 1/ 3 ( 10 số hạng)

Tương tự :( 1/41 +1/42+  1/43 +1/44+ 1/45 +1/46+  1/47 +1/48 + 1/49 +1/50)<1/4 và

(1/51 +1/52+  1/53 +1/54+ 1/55 +1/56+  1/57 +1/58 + 1/59 +1/60)< 1/5

(1/3 + 1/4 + 1/5 ) < 4/5 ( dpcm)

Ưu ái lắm nha!