奧數(速算+巧算)

1)1/(1+2) + 1/(1+2+3) + ... + 1/(1+2+...+99) + 1/(1+2+...+100)= 2/(2x3) + 2/(3x4) + ... + 2/(99x100) + 2/(100x101)= 2 ( (1/2 - 1/3) + (1/3 - 1/4) + ... + (1/99 - 1/100) + (1/100 - 1/101) )= 2 (1/2 - 1/101)= 99 / 101
2) (1+1/3) - 7/12 + 9/20 - 11/30 - 13/42 - 15/56 + 17/72 - 19/90= (1+1/3) - 7/12 + 9/20 - 11/30 - 13/42 + 15/56 + 17/72 - 19/90 - 2(15/56)= (1+1/3) - (3+4)/(3x4) + (4+5)/(4x5) - (5+6)/(5x6) + (6+7)/(6x7)
- (7+8)/(7x8) + (8+9)/(8x9) - (9+10)/(9x10) - 2(15/56)= (1+1/3) - (1/3 + 1/4) + (1/4 + 1/5) - (1/5 + 1/6) + (1/6 + 1/7)
- (1/7 + 1/8) + (1/8 + 1/9) - (1/9 + 1/10) - 2(15/56)= 1 + 1/10 - 2(15/56)= 79 / 140
3)1/(1x2x3) + 1/(2x3x4) + ... + 1/(20x21x22)= (1/2)(3 - 1)/(1x2x3) + (1/2)(4 - 2)/(2x3x4) + ... + (1/2)(22 - 20)/(20x21x22)= (1/2) [ 1/(1x2) - 1/(2x3) + 1/(2x3) - 1/(3x4) + ... + 1/(20x21) - 1/(21x22) ] = (1/2) [ 1/(1x2) - 1/(21x22) ]= (1/2) (1/2 - 1/462)= 115 / 462
4) (1 - 1/11²) x (1 - 1/12²) x ... x (1 - 1/98²) x (1 - 1/99²)= (1 - 1/11)(1 + 1/11) x (1 - 1/12)(1 + 1/12) x ...
x (1 - 1/98)(1 + 1/98) x (1 - 1/99)(1 + 1/99)= (10/11)(12/11) x (11/12)(13/12) x ... x (97/98)(99/98) x (98/99)(100/99)= (10/11) (100/99)= 1000 / 1089
5)1 x 2 + 2 x 3 + 3 x 4 + ... + 28 x 29 + 29 x 30= (1 x 2) (3 - 0)/3 + (2 x 3) (4 - 1)/3 + (3 x 4) (5 - 2)/3 + ...
+ (28 x 29) (30 - 27)/3 + (29 x 30) (31 - 28)/3
= (1 x 2 x 3 - 0 x 1 x 2)/3 + (2 x 3 x 4 - 1 x 2 x 3)/3 + (3 x 4 x 5 - 2 x 3 x 4)/3
+ ... + (28 x 29 x 30 - 27 x 28 x 29)/3 + (29 x 30 x 31 - 28 x 29 x 30)/3= (- 0 x 1 x 2)/3 + (29 x 30 x 31)/3= (29 x 30 x 31)/3= 8990
6) 135792468 / (135792468² - 135792467 x 135792469)令 n = 135792468 , 原式= n / (n² - (n-1)(n+1) )= n / (n² - (n² - 1))= n = 135792468
7)1² + 2² + 3² + ... + n²= n (n+1) (2n+1) / 6 這是十分常見的公式。
8) 1/2 + 1/6 + 1/12 + 1/20 + 1/30 + 1/42 + 1/56 + 1/72 + 1/90= 1/(1x2) + 1/(2x3) + 1/(3x4) + 1/(4x5) + 1/(5x6) + 1/(6x7) + 1/(7x8) + 1/(8x9)
+ 1/(9x10)= (1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) + ... + (1/8 - 1/9) + (1/9 - 1/10)= 1 - 1/10= 9/10

2011-10-23 23:33:34 補充：
修正 :

2)

(1+1/3) - 7/12 + 9/20 - 11/30 - 13/42 - 15/56 + 17/72 - 19/90

= (1+1/3) - 7/12 + 9/20 - 11/30 + 13/42 - 15/56 + 17/72 - 19/90 - 2(13/42)

= (1+1/3) - (3+4)/(3x4) + (4+5)/(4x5) - (5+6)/(5x6) + (6+7)/(6x7)
- (7+8)/(7x8) + (8+9)/(8x9) - (9+10)/(9x10) - 2(13/42)

2011-10-23 23:33:39 補充：
= (1+1/3) - (1/3 + 1/4) + (1/4 + 1/5) - (1/5 + 1/6) + (1/6 + 1/7)
- (1/7 + 1/8) + (1/8 + 1/9) - (1/9 + 1/10) - 2(13/42)

= 1 - 1/10 - 2(13/42)

= 59 / 210

2011-10-24 00:40:40 補充：
打得好辛苦架 , 記住選我呀 ~

2011-10-24 23:10:30 補充：
你的計算第二括號多了 1 ,

=(1^2+2^2+3^2+4^2+...+29^2+30^2-1^2) - (1+2+3+4+...+29+30)

應為

=(1^2+2^2+3^2+4^2+...+29^2+30^2-1^2) - (2+3+4+...+29+30)

2011-10-24 23:11:53 補充：
第 3 題同第 1 , 8 題差不多。

第 3 題你怎想到的？