已知数列{An}满足A1=1,点(An,A(n+1))在直线y=2x+1上.若数列{bn}满足b1=A1,bn/An=1/A1+1/A2+1/A3+...+1/A(n-1),求证：(1+b1)(1+b2)(1+b3)...(1+bn)＜10/3*b1*b2*b3...bn

Question

已知数列{An}满足A1=1,点(An,A(n+1))在直线y=2x+1上.
若数列{bn}满足b1=A1,bn/An=1/A1+1/A2+1/A3+...+1/A(n-1),
求证：(1+b1)(1+b2)(1+b3)...(1+bn)＜10/3* b1* b2 * b3 ...bn

Answer 1

由题意知A(n+1)=2A(n)+1,即A(n+1)+1=2[A(n)+1],故A(n)=2^n-1..
又bn/An=1/A1+1/A2+1/A3+...+1/A(n-1),所以1+bn=A(n)[1/A(1)+...+1/A(n)],要证结论等价于证
A(1)A(2)[1/A(1)+1/A(2)]...A(n)[1/A(1)+...+1/A(n)]