一道数学数列题设两个数列{An},{Bn}满足Bn=（A1+A2+A3+……+nAn)/(1+2+3+……+),若{Bn}为等差数列,求证{An}也是等差数列.那个，题目上是“若{Bn}为等差数列，求证{An}也是等差数列”要是“若{an}为等差数列，求证数列{bn}也是等差数列”就简单了不过还是谢谢你 A1是{An}的第一项，不是A的一次方

数列{an}，{bn}满足 bn=(1*a1+2*a2+3*a3…+nan)/(1+2+3+…+n)
若{an}为等差数列，求证数列{bn}也是等差数列
证明：若{an}为等差数列,设公差为d,则
b[n]={(1+2+……+n)*a1+[1*2+2*3+……+(n-1)n]d}/(1+2+……+n)
=a1+(1^2+2^2+3^2+……+n^2)d-(1+2+……+n)d/(1+2+……+n)
=a1+[n(n+1)(2n+1)d/6-n(n+1)d/2]/[n(n+1)/2]
=a1+(2n+1)d/3-d
=a1+(2n-2)d
=a1+(n-1)*2d
显然，b[n]是以a1为首项，2d为公差的等差数列。命题得证。

B为常数列
A1等于B1
A为公差为B的常数值的等差数列
有很多方法
仔细想想
别让别人做你的大脑哦

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……