證明題 (20分)

Prove that for any positive integers n
1*n + 2*(n－1) + 3*(n－2) + ... + (n－1)*2 + n*1=n(n+1)(n+2)/6
Sol
1*n + 2*(n－1) + 3*(n－2) + ... + (n－1)*2 + n*1
=Σ(k=1 to n)_[k*(n+1－k)]
=Σ(k=1 to n)_[(n+1)k－k^2]
=(n+1)Σ(k=1 to n)_k－Σ(k=1to n)_k^2
=(n+1)*[n(n+1)/2]－n(n+1)(2n+1)/6
=[n(n+1)/6]*[3(n+1)－(2n+1)]
=[n(n+1)/6]*(n+2)
=n(n+1)(n+2)/6
儘量不要用數學歸納法