两个数列{an}和{bn}满足bn=(a1+2a2+3a3…+nan)/(1+2+3+…+n)(n∈N*).1+2+3+…n=(n(n+1))/2(1)若数列{bn}是等差数列,求证:{an}也是等差数列;(2)若数列{an}是等差数列,求证:{bn}也是等差数列.
问题描述:
两个数列{an}和{bn}满足bn=(a1+2a2+3a3…+nan)/(1+2+3+…+n)(n∈N*).1+2+3+…n=(n(n+1))/2
(1)若数列{bn}是等差数列,求证:{an}也是等差数列;
(2)若数列{an}是等差数列,求证:{bn}也是等差数列.
答
bn=(a1+2a2+3a3…+nan)/(1+2+3+…+n) ;
1+2+3+…n=(n(n+1))/2;
bn=2(a1+2a2+3a3…+nan)/n*(n+1)
b1=a1
b2=2(a1+2a2)/2*3=(a1+2a2)/3,b2-b1=2/3*(a2-a1)=2/3 *公差q;
b3=2(a1+2a2+3a3)/12=(a1+2a2+3a3)/6,b3-b2=2/3 *公差q;
所以,如果
若数列{an}是等差数列,假设公差为q,{bn}也是等差数列,公差为2/3 *公差q;
同理可证,如数列{bn}是等差数列,{an}也是等差数列