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)
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