Question

如何计算下题： 1×2×3+2×3×4+3×4×5+4×5×6+5×6×7+6×7×8+7×8×9+……+n×(n+1)

Answer 1

(n-1)(n+1)=n^2-1 则1*2*3=2^3-2 2*3*4=3^3-3 .则原式=2^3+3^3+4^3+...+(n+1)^3-2-3-4-...(n+1) 1^3+2^3+...+n^3=[n(n+1)/2]^2 所以2^3+3^3+4^3+...+(n+1)^3=[(n+1)(n+2)/2]^2-1 2+3+4+...+(n+1)=(2+n+1)n/2 所以原式=[(n+1)(n+2)/2]^2-1-(n+3)n/2