an,（bn)^2,a(n+1)成等差数列,（bn）^2,a(n+1),(b(n+1))^2成等比数列,证：（bn）为等差数列

题目

an,（bn)^2,a(n+1)成等差数列,（bn）^2,a(n+1),(b(n+1))^2成等比数列,证：（bn）为等差数列

答案

由题an,（bn)^2,a(n+1)成等差数列2(bn)^2=an+a(n+1)--①由（bn）^2,a(n+1),(b(n+1))^2成等比数列(a(n+1))^2=[bnb(n+1)]^2∴a(n+1)=bnb(n+1)在①中有2(bn)^2=bn(b(n-1)+b(n+1))即2bn=b(n-1)+b(n+1)故{bn}为等差数列...