数列证明题一题设数列{An}满足：A1=1,且当n∈N*时,An^3+An^2×[1－A(n+1)]+1=A(n+1）求证：数列{_数学解答

数列证明题一题
设数列{An}满足：A1=1,且当n∈N*时,
An^3+An^2×[1－A(n+1)]+1=A(n+1）
求证：数列{An}是递增数列.

人气：177 ℃　时间：2020-09-22 21:51:18

解答

证明：由题
A(n+1)=[An^3+An^2+1]/[An^2+1]
=[An^3+An+An^2+1-An]/[An^2+1]
=An+1-An/[An^2+1]
故A(n+1)-A(n)
=1-An/[An^2+1]
=[An^2+1-An]/[An^2+1]
=[(An-1/2)^2+3/4]/[An^2+1]
>=0
故A(n+1)>=A(n)
故数列{An}是递增数列