Question

[Kiên NT]

chứng tỏ rằng tổng của các phân số sau đây lớn hơn 1/2 : S= 1/5+1/51+1/52+...+1/98+1/99

giúp mik với nha mik sắp nộp rùi

Answer 1

[Ilovehoc24]

ta có 1/50>1/100

1/51>1/100

1/52>1/100

................

1/99>1/100

suy ra S=1/50+1/51+1/52+..........+1/99>1/100x50=1/2

suy ra S=1/2

Answer 2

[qwerty]

Ta có: 

A = 1/2-1/3+1/4-1/5+1/6-1/7+ ..... +1/98-1/99 

=> -A = -1/2+1/3-1/4+1/5-1/6+1/7+ ..... -1/98+1/99 

=> -A = 1/2+1/3+1/4+1/5+ ... +1/98+1/99 - 2.(1/2+1/4+1/6+...+1/98) 

=> -A = 1/2+1/3+1/4+1/5+ ... +1/98+1/99 -(1+1/2+1/3+1/4+...+1/49) 

=> -A = -1+1/50+1/51+1/52+ ... +1/99 


Đặt: B = 1/50+1/51+1/52+ ... +1/99 

=> B = (1/50 +1/51+...+1/59) +(1/60+1/61+...+1/69) +(1/70+1/71+...+1/79) +(1/80+1/81+...+1/89) +(1/90+1/91+...+1/99) 

Do đó: 

10.(1/59)+10.(1/69)+10.(1/79) +10.(1/89)+10.(1/99) < B < 10.(1/50)+10.(1/60)+10.(1/70) +10.(1/80)+10.(1/90) 

=> 10.(1/60)+10.(1/70)+10.(1/80) +10.(1/90)+10.(1/100) < B < 10.(1/50)+10.(1/60)+10.(1/70) +10.(1/80)+10.(1/90) 

=> 1/6 +1/7 +1/8 +1/9 +1/10 < B < 1/5 +1/6 +1/7 +1/8 +1/9 

=> 0,6456 < B < 0,7456 

=> 3/5 < B < 4/5 

=> -2/5 < -1+B < -1/5 

=> -2/5 < -A < -1/5 

=> 1/5 < A <2/5