1/1x2x3x4 + 1/2x3x4x5 + 1/3x4x5x6 + ...........+
1/97x98x99x100 =? :-?
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bubupin

2007-08-12 23:34
1/1*2*3*4=1/(1*4)*(2*3)=(1/2)(1/1*4-1/2*3)

1/1*2*3*4 + 1/2*3*4*5 + 1/3*4*5*6 +......+1/97*98*99*100
=(1/2)(1/1*4-1/2*3+1/2*5-1/3*4+....+1/97*100-1/98*99)
=(1/2)[1/1*4+1/2*5+...+1/97*100-(1/2*3+1/3*4+...+1/98*99)]

(1)
1/1*4+1/2*5+1/3*6+......+1/97*100
=(1/3)*(1-1/4+1/2-1/5+1/3-1/6+........+1/97-1/100)
=(1/3)*[(1+1/2+1/3+1/4+1/5+....+1/97)-(1/4+1/5+....+1/97+1/98+1/99+1/100)]
=(1/3)*(1+1/2+1/3-1/98-1/99-1/100)
(2)
1/2*3+1/3*4+1/4*5---+1/98*99
=1/2-1/3+1/3-1/4+1/4-1/5+.....1/98-1/99=1/2-1/99
因此
1/1*2*3*4 + 1/2*3*4*5 + ...........+1/97*98*99*100
=(1/2)[(1/3)*(1+1/2+1/3-1/98-1/99-1/100)-(1/2-1/99)]
=(1/2)(1/3+1/6+1/9-1/294-1/297-1/300-1/2+1/99)
=(1/2)(1/9+1/99-1/294-1/297-1/300)
=(1/2)(35/297-1/294-1/300)