Question

【高考数学】21、已知数列 ，且 。求证： 为等差数列的充要条件是 为等差数列。

Answer 1

必要性：设an公差为d，则
bn=(a1+2a2+3a3+…+nan)/(1+2+3+…+n)
=2(a1+2a2+3a3+…+nan)/n(n+1)
=2(a1+2(a1+d)+3(a1+2d)+…+n(a1+(n-1)d)/n(n+1)
=2{(a1+2a1+3a1+…+na1)+[1*2+2*3+3*4+…(n-1)n]d}/n(n+1)
=2{(n(n+1)a1/2)+[1*2+2*3+3*4+…(n-1)n]d}/n(n+1)
={(n(n+1)a1)+2[1*2+2*3+3*4+…(n-1)n]d}/n(n+1)
=a1+2[1*2+2*3+3*4+…+(n-1)n]d/n(n+1)
=a1+2[1+2+3+…+n-1+1^2+2^2+3^2+…+(n-1)^2]d/n(n+1)
=a1+2(n-1)n(n+1)d/3n(n+1)
=a1+(n-1)2d/3
即是bn是以a1为首数，2d/3为公差的等差数列 同理可证必要性：当bn为等差数列时，an为等差数列 所以是数列{bn}等差数列冲要条件{an}是等差数列