求1/x(x+4) + 1/(x+4)(x+8) + 1/(x+8)(x+12) + 1/(x+12)(x+16)的值

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求1/x(x+4) + 1/(x+4)(x+8) + 1/(x+8)(x+12) + 1/(x+12)(x+16)的值

求1/x(x+4) + 1/(x+4)(x+8) + 1/(x+8)(x+12) + 1/(x+12)(x+16)的值
求1/x(x+4) + 1/(x+4)(x+8) + 1/(x+8)(x+12) + 1/(x+12)(x+16)的值

求1/x(x+4) + 1/(x+4)(x+8) + 1/(x+8)(x+12) + 1/(x+12)(x+16)的值
1/x(x+4) + 1/(x+4)(x+8) + 1/(x+8)(x+12) + 1/(x+12)(x+16)
=(1/4)[1/x-1/(x+4)+1/(x+4)-1/(x+8)+1/(x+8)-1/(x+12)+1/(x+12)-1/(x+16)]
=(1/4)[1/x-1/(x+16)]
=(1/4)(x+16-x)/[x(x+16)]
=4/[x(x+16)]

将乘积变为差的形式
由于有1/x(x+4)=1/4(1/x-1/(x+4))
所以原式=1/4(1/x-1/(x+16))=4/x(x+16)