观察下列等式1/1×2=1-1/2,1/2×3=1/2-1/3,1/3×4=1/3-1/4将以上三个等式两边分别相加得:

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  • 1/2×1+1/2×3+1/3×4=1-1/2+1/2-1/3+1/3-1/4=1-1/4=3/4

    (1)猜想并写出1/n(n+1)=1/n - 1/(n+1)

    (2)直接写出下列各式的计算结果:

    1、1/1×2+1/2×3+1/3×4…+1/2006×2007=1-1/2007=2006/2007

    2、1/1×2+1/2×3+1/3×4+…1/n(n+1)=1- 1/(n+1)

    (3)探究并计算:

    1/2×4+1/4×6+1/6×8+…1/2006×2008

    =½x(1/2-1/4)+½x(1/4-1/6)+……+½x(1/2006-1/2008)

    =½x(1/2-1/4+1/4-1/6+……+1/2006-1/2008)

    =½x(1/2-1/2008)

    =½x(1003/2008)

    =1003/4016