你题目写错了,{Bn}的表达式应该是
Bn=（A1+2A2+3A3+……+nAn)/(1+2+3+……+n)
那啥,第n+1项我直接用B(n+1)来表示,你应该能看懂
设Bn公差为d
Bn=（A1+2A2+3A3+……+nAn)/(1+2+3+……+n)
=2(A1+2A2+3A3+……+nAn)/(n(n+1))
=2(A1+2A2+3A3+……+nAn)(n+2)/(n(n+1)(n+2))
B(n+1)=（A1+2A2+3A3+……+nAn+(n+1)A(n+1))/(1+2+3+……+n+(n+1))
=2（A1+2A2+3A3+……+nAn+(n+1)A(n+1))/((n+1)(n+2))
=2（A1+2A2+3A3+……+nAn+(n+1)A(n+1))n/(n(n+1)(n+2))
由Bn+d=B(n+1),得
2n(A1+2A2+3A3+……+nAn)+4(A1+2A2+3A3+……+nAn)+n(n+1)(n+2)d=
2n(A1+2A2+3A3+……+nAn)+2n(n+1)A(n+1)
4(A1+2A2+3A3+……+nAn)+n(n+1)(n+2)d=2n(n+1)A(n+1)……1式
用n-1代换n,得
4(A1+2A2+3A3+……+(n-1)A(n-1))+(n-1)n(n+1)d=2n(n-1)An……2式
1式-2式,得
4nAn+3n(n+1)d=2n(n+1)A(n+1)-2n(n-1)An
2n(n+1)An+3n(n+1)d=2n(n+1)A(n+1)
An+1.5d=A(n+1)
得证……