1/1×2×3+1/2×3×4+1/3×4×5……+1/20×21×22

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1/1×2×3+1/2×3×4+1/3×4×5……+1/20×21×22
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1/1×2×3+1/2×3×4+1/3×4×5……+1/20×21×22
1/1×2×3+1/2×3×4+1/3×4×5……+1/20×21×22

1/1×2×3+1/2×3×4+1/3×4×5……+1/20×21×22
1/n(n+1)(n+2)=(1/n(n+1)-1/(n+1)(n+2))/2
1/1×2×3+1/2×3×4+1/3×4×5……+1/20×21×22
=(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
=(1/1*2-1/21*22)*1/2
=(1/2-1/462)/2
=115/462