已知数列{An}的首项A1=3/5,A[n+1]=3An/(2An+1),n=1,2...(1)求{An}的通项公式;(2)证明:对任意的x>0,An>=[1/(1+x)]-{1/[(1+x)^2]}*[2/(3^n)-x],n=1,2...;(3)证明:(A1+A2+...+An)>[(n^2)/(n+1)]
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![已知数列{An}的首项A1=3/5,A[n+1]=3An/(2An+1),n=1,2...(1)求{An}的通项公式;(2)证明:对任意的x>0,An>=[1/(1+x)]-{1/[(1+x)^2]}*[2/(3^n)-x],n=1,2...;(3)证明:(A1+A2+...+An)>[(n^2)/(n+1)]](/uploads/image/z/5128283-11-3.jpg?t=%E5%B7%B2%E7%9F%A5%E6%95%B0%E5%88%97%7BAn%7D%E7%9A%84%E9%A6%96%E9%A1%B9A1%3D3%2F5%2CA%5Bn%2B1%5D%3D3An%2F%282An%2B1%29%2Cn%3D1%2C2...%281%29%E6%B1%82%7BAn%7D%E7%9A%84%E9%80%9A%E9%A1%B9%E5%85%AC%E5%BC%8F%3B%282%29%E8%AF%81%E6%98%8E%3A%E5%AF%B9%E4%BB%BB%E6%84%8F%E7%9A%84x%3E0%2CAn%3E%3D%5B1%2F%281%2Bx%29%5D-%7B1%2F%5B%281%2Bx%29%5E2%5D%7D%2A%5B2%2F%283%5En%29-x%5D%2Cn%3D1%2C2...%3B%283%29%E8%AF%81%E6%98%8E%3A%28A1%2BA2%2B...%2BAn%29%3E%5B%28n%5E2%29%2F%28n%2B1%29%5D)
已知数列{An}的首项A1=3/5,A[n+1]=3An/(2An+1),n=1,2...(1)求{An}的通项公式;(2)证明:对任意的x>0,An>=[1/(1+x)]-{1/[(1+x)^2]}*[2/(3^n)-x],n=1,2...;(3)证明:(A1+A2+...+An)>[(n^2)/(n+1)]
已知数列{An}的首项A1=3/5,A[n+1]=3An/(2An+1),n=1,2...
(1)求{An}的通项公式;
(2)证明:对任意的x>0,An>=[1/(1+x)]-{1/[(1+x)^2]}*[2/(3^n)-x],n=1,2...;
(3)证明:(A1+A2+...+An)>[(n^2)/(n+1)]
已知数列{An}的首项A1=3/5,A[n+1]=3An/(2An+1),n=1,2...(1)求{An}的通项公式;(2)证明:对任意的x>0,An>=[1/(1+x)]-{1/[(1+x)^2]}*[2/(3^n)-x],n=1,2...;(3)证明:(A1+A2+...+An)>[(n^2)/(n+1)]
由已知 A[n+1]=3An/(2An+1)
1/A[n+1]=(2An+1)/3An=1/(3An)+2/3
设1/An=Bn 则B[n+1]=1/3*Bn+2/3
等式两边各-1 B[n+1]-1=1/3*Bn-1/3=1/3*(Bn-1)
B[n+1]-1是等比数列 首项为B1-1=1/A1-1=2-3 公比为1/3
所以 Bn-1=2/3*(1/3)^(n-1)=2*(1/3)^n
Bn=2*(1/3)^n+1
An=1/Bn=1/[2*(1/3)^n+1]