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3x4 4x5 5x6 20x21

3x4+4x5+5x6+........19x20+20x21 =3^2+3+4^2+4+...+20^2+20 =3^2+4^2+...+20^2+3+4+..+20 1^2+2^2+3^2+……+n^2=n(n+1)(2n+1)/6 1^2+2^2+...+20^2=20*21*41/6 1^2+2^2=5 所以: 3x4+4x5+5x6+........19x20+20x21 =3^2+3+4^2+4+...+20^2+20 =3^2+4...

式子可以分为1/n-1/(n+1)的形式,之后约分即可

1/1×2×3=1/2×(1/1×2-1/2×3) 其他以此类推 所以原式=1/2×(1/1×2-1/2×3+1/2×3-1/3×4+……+1/20×21-1/21×22) =1/2×(1/1×2-1/21×22) =115/462

1/(1*2*3)+1/(2*3*4)+1/(3*4*5)+.........1/(20*21*22) =1/2[1/1*2-1/2*3]+1/2[1/2*3-1/3*4]+1/2[1/3*4-1/4*5]+...+1/2[1/20*21-1/21*22] =1/2[1/1*2-1/2*3+1/2*3-1/3*4+1/3*4-1/4*5+...+1/20*21-1/21*22] =1/2[1/2-1/21*22] =1/2*115/231 =115/462

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