简便计算：1/35+1/63+1/99+1/143

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简便计算：1/35+1/63+1/99+1/143

1/35+1/63+1/99+1/143
=1/(5*7)+1/(7*9)+1/(9*11)+1/(11*13)
=(1/2)*[(1/5)-(1/7)]+(1/2)*[(1/7)-(1/9)]+(1/2)*[(1/9)-(1/11)]+(1/2)*[(1/11)-(1/13)]
=(1/2)*[(1/5)-(1/7)+(1/7)-(1/9)+(1/9)-(1/11)+(1/11)-(1/13)]
=(1/2)*[(1/5)-(1/13)]
=(1/2)*(8/65)
=4/65.

1/35+1/63+1/99+1/143
=1/(5*7)+1/(7*9)+1/(9*11)+1/(11*13)
=(1/2)*[(1/5)-(1/7)]+(1/2)*[(1/7)-(1/9)]+(1/2)*[(1/9)-(1/11)]+(1/2)*[(1/11)-(1/13)]
=(1/2)*[(1/5)-(1/7)+(1/7)-(1/9)+(1/9)-(1/11)+(1/11)-(1/13)]
=(1/2)*[(1/5)-(1/13)]
=(1/2)*(8/65)
=4/65.

裂项法
原式=1/(5*7)+1/(7*9)+1/(9*11)+1/(11*13)
=1/2*(1/5-1/7)+1/2*(1/7-1/9)+1/2*(1/9-1/11)+1/2(1/11-1/13)
=1/2*(1/5-1/7+1/7-1/9+1/9-1/11+1/11-1/13)
=1/2*(1/5-1/13)
=1/2*8/65
=4/65

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