求和:1X2X3+2X3X4+3X4X5+...+98X99X100
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求和:1X2X3+2X3X4+3X4X5+...+98X99X100
![求和:1X2X3+2X3X4+3X4X5+...+98X99X100](/uploads/image/z/16958654-62-4.jpg?t=%E6%B1%82%E5%92%8C%EF%BC%9A1X2X3%2B2X3X4%2B3X4X5%2B...%2B98X99X100)
因为(n-1)n(n+1)=n(n²-1)=n³-n
∴原式=2³-2+3³-3+4³-4+……+99³-99
=2³+3³+4³+……+99³-(2+3+4+……+99)
=1³+2³+3³+……+99³-(1³+2+3+……+99)
【这里要知道:连续自然数的立方和等于他们和的平方,即
1³+2³+3³+……+n³=(1+2+3+……+n)²=[n(n+1)/2]²】
=[99×(99+1)/2]²-99×(99+1)/2
=99×100/2×(99×100/2-1)
=99×50×(99×50-1)
=24497550
【本题总结】
1×2×3+2×3×4+……+(n-1)n(n+1)
=1³+2³+3³+……+n³-(1+2+3+……+n)
=[n(n+1)/2]²-n(n+1)/2
=[n(n+1)/2][n(n+1)/2-1]
=1/4×{n(n+1)[n(n+1)-2]}
∴原式=2³-2+3³-3+4³-4+……+99³-99
=2³+3³+4³+……+99³-(2+3+4+……+99)
=1³+2³+3³+……+99³-(1³+2+3+……+99)
【这里要知道:连续自然数的立方和等于他们和的平方,即
1³+2³+3³+……+n³=(1+2+3+……+n)²=[n(n+1)/2]²】
=[99×(99+1)/2]²-99×(99+1)/2
=99×100/2×(99×100/2-1)
=99×50×(99×50-1)
=24497550
【本题总结】
1×2×3+2×3×4+……+(n-1)n(n+1)
=1³+2³+3³+……+n³-(1+2+3+……+n)
=[n(n+1)/2]²-n(n+1)/2
=[n(n+1)/2][n(n+1)/2-1]
=1/4×{n(n+1)[n(n+1)-2]}
1/1x2x3+1/2x3x4+1/3x4x5+.1/98x99x100=
1/1x2x3+1/2x3x4+1/3x4x5
1x2X3+2x3X4+3x4X5+…+7X8X9=?
1x2x3+2x3x4+3x4x5+…+8x9x10
1x2x3+2x3x4+3x4x5+...+7x8x9=,
1x2x3+2x3x4+3x4x5+.+10x11x12
有理数:1.计算:1/1x2x3+1/2x3x4+1/3x4x5+...+1/98x99x1002.求式子:|x+1|+
1/1x2x3+1/2x3x4+1/3x4x5+1/4x5x6+.+1/48x49x50=
1/1x2x3+1/2x3x4+1/3x4x5+.+1/11x12x13=
1/1x2x3+1/2x3x4+1/3x4x5+.+1/9x10x11=
请1/1x2x3+1/2x3x4+1/3x4x5+1/4x5x6=
1/(1x2x3)+1/(2x3x4)+1/(3x4x5)+.1/(20x21x22)=?