设数列{An}和{bn}满足A1=1/2,2nA(n+1)=(n+1)An,且Bn=ln(1+An)+1/2(An)2,n属于N+(1):求A2,A3,A4,并求数列{An}的通项公式(2):对一切n属于N+,证明2/(An+2)小于An/Bn成立
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![设数列{An}和{bn}满足A1=1/2,2nA(n+1)=(n+1)An,且Bn=ln(1+An)+1/2(An)2,n属于N+(1):求A2,A3,A4,并求数列{An}的通项公式(2):对一切n属于N+,证明2/(An+2)小于An/Bn成立](/uploads/image/z/3007285-61-5.jpg?t=%E8%AE%BE%E6%95%B0%E5%88%97%7BAn%7D%E5%92%8C%7Bbn%7D%E6%BB%A1%E8%B6%B3A1%3D1%2F2%2C2nA%28n%2B1%29%3D%28n%2B1%29An%2C%E4%B8%94Bn%3Dln%281%2BAn%29%2B1%2F2%28An%292%2Cn%E5%B1%9E%E4%BA%8EN%2B%EF%BC%881%EF%BC%89%EF%BC%9A%E6%B1%82A2%2CA3%2CA4%2C%E5%B9%B6%E6%B1%82%E6%95%B0%E5%88%97%7BAn%7D%E7%9A%84%E9%80%9A%E9%A1%B9%E5%85%AC%E5%BC%8F%EF%BC%882%EF%BC%89%EF%BC%9A%E5%AF%B9%E4%B8%80%E5%88%87n%E5%B1%9E%E4%BA%8EN%2B%2C%E8%AF%81%E6%98%8E2%2F%EF%BC%88An%2B2%29%E5%B0%8F%E4%BA%8EAn%2FBn%E6%88%90%E7%AB%8B)
设数列{An}和{bn}满足A1=1/2,2nA(n+1)=(n+1)An,且Bn=ln(1+An)+1/2(An)2,n属于N+(1):求A2,A3,A4,并求数列{An}的通项公式(2):对一切n属于N+,证明2/(An+2)小于An/Bn成立
设数列{An}和{bn}满足A1=1/2,2nA(n+1)=(n+1)An,且Bn=ln(1+An)+1/2(An)2,n属于N+(1):求A2,A3,A4,并求数列{An}的通项公式(2):对一切n属于N+,证明2/(An+2)小于An/Bn成立
设数列{An}和{bn}满足A1=1/2,2nA(n+1)=(n+1)An,且Bn=ln(1+An)+1/2(An)2,n属于N+(1):求A2,A3,A4,并求数列{An}的通项公式(2):对一切n属于N+,证明2/(An+2)小于An/Bn成立
(1)A1=1/2,2nA(n+1)=(n+1)An,
∴A/(n+1)=(1/2)An/n=…=1/2^n,
∴An=n/2^n.A2=1/2,A3=3/8,A4=1/4.
(2)An(An+2)-2Bn
=2[An-ln(1+An)]>0,
An,Bn>0,
∴原式成立.
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