數學知識交流---整數部份(2)

1)12/34 + 56/78 + 90/12 + 34/56 + 78/90= (12/34 + 34/56) + (56/78 + 78/90) + 90/12利用 A.M. > G.M. :> 2√[(12/34)(34/56)] + 2√[(56/78)(78/90)] + 90/12= 2√(12/56) + 2√(56/90) + 90/12= 2√[(3/7)(4/8)] + 2√[(7/9)(8/10)] + 90/12> 2(3/7) + 2(7/9) + 90/12 = 6/7 + (1+5/9) + 7.5= 9.9....推測整數部份極可能是 10 。而 12/34 + 56/78 + 90/12 + 34/56 + 78/90 - 10= (12/34 - 1) + (56/78 - 1) + (90/12 - 6) + (34/56 - 1) + (78/90 - 1)= - 22/34 - 22/78 + 3/2 - 22/56 - 12/90≈ - 22/33 - 22/77 + 3/2 - 22/55 - 2/15≈ 3/2 - (2/3 + 2/7 + 2/5 + 2/15)≈ 3/2 - 2(1/3 + 1/7 + 1/5 + 1/15)≈ 3/2 - 2(26/35)≈ 1/70> 0故 12/34 + 56/78 + 90/12 + 34/56 + 78/90 的整數部份的確為10。
2)1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/10= (1/2 + 1/3 + 1/6) + (1/4 + 1/5) + (1/7 + 1/8 + 1/9 + 1/10)< 1 + (1/4 + 1/4) + (1/7 + 1/7 + 1/7 + 1/7)= 1 + 1/2 + 3/7 < 2故其整數部份是 1。

2011-06-02 23:40:46 補充：
修正

2)

1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/10

= (1/2 + 1/3 + 1/6) + (1/4 + 1/5 + 1/10) + (1/7 + 1/8 + 1/9)

= (1) + (0.55) + (1/7 + 1/8 + 1/9)

< 1.55 + (1/7 + 1/7 + 1/7)

= 1.55 + 3/7

= 1.55 + 0.42857... < 2

故其整數部份是 1。

2011-06-03 13:23:52 補充：
1)'s solution proves the integral part of

(12/34 + 56/78 + 90/12 + 34/56 + 78/90) ≈ 10 + 1/70 < 11

2011-06-03 13:40:45 補充：
這樣會清楚些 :

(12/34 + 56/78 + 90/12 + 34/56 + 78/90)

= (1 - 22/34) + (1 - 22/78) + 7.5 + (1 - 22/56) + (1 - 12/90)

= 11.5 - (22/34 + 22/78 + 22/56 + 2/15)

>≈ 11.5 - (22/33 + 22/77 + 22/55 + 2/15)

= 11.5 - 2(1/3 + 1/7 + 1/5 + 1/15)

= 11.5 - 2(26/35)

= 10 + 1/70

(1)'s solution only proves the integral part of (12/34+56/78+90/12+34/56+78/90) ≧10, but why is it 10 instead of 11, 12, 13, 14, ...

(1) the result is: 310783 / 30940 = 10 + 1383 / 30940
so the 整數部分 is 10.

2011-06-02 20:38:48 補充：
(2) the result is: 4861 / 2520 = 1 + 2341 / 2520
so the 整數部分 is 1.