六年级奥数巧算题 1/(2*3*4)+1/(3*4*5)+1/(4*5*6)+.+1/(8*9*10)

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六年级奥数巧算题 1/(2*3*4)+1/(3*4*5)+1/(4*5*6)+.+1/(8*9*10)
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六年级奥数巧算题 1/(2*3*4)+1/(3*4*5)+1/(4*5*6)+.+1/(8*9*10)
六年级奥数巧算题 1/(2*3*4)+1/(3*4*5)+1/(4*5*6)+.+1/(8*9*10)

六年级奥数巧算题 1/(2*3*4)+1/(3*4*5)+1/(4*5*6)+.+1/(8*9*10)
1/(2*3*4)+1/(3*4*5)+1/(4*5*6)+.+1/(8*9*10)
=1/2*[1/2-2/3+1/4+1/3-2/4+1/5+1/4-2/5+1/6...+1/8-2/9+1/10]
=1/2*[1/2-1/3-1/9+1/10]
=7/90

1/(2*3*4)+1/(3*4*5)+1/(4*5*6)+。。。。+1/(8*9*10)
=1/2×【1/(2*3)-1/(3*4)+1/(3*4)-1/(4*5)+ --- +1/(8*9)-1/(9*10)】
=1/2×【1/(2*3)-1/(9*10)】
=1/2×7/45
7/90

1/[(n-1)n(n-2)]=1/2*[1/(n-1)-2/n+1/(n+1)]
1/(2*3*4)+1/(3*4*5)+1/(4*5*6)+。。。。+1/(8*9*10)
=1/2*[1/2-2/3+1/4+1/3-2/4+1/5+1/4-2/5+1/6...+1/8-2/9+1/10]
=1/2*[1/2-1/3-1/9+1/10] (中间全部约掉)
=7/90

(1/2*3-1/3*4+。。。1/8*9-1/9*10)/2=(1/6-1/90)/2=7/18