数列{An},A1=4/3,A2=13/9,当n>=3时,An-An-1=1/3(An-1-An-2),求{An}的通项公式,前n项和Sn是多少
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![数列{An},A1=4/3,A2=13/9,当n>=3时,An-An-1=1/3(An-1-An-2),求{An}的通项公式,前n项和Sn是多少](/uploads/image/z/13331997-45-7.jpg?t=%E6%95%B0%E5%88%97%7BAn%7D%2CA1%3D4%2F3%2CA2%3D13%2F9%2C%E5%BD%93n%3E%3D3%E6%97%B6%2CAn-An-1%3D1%2F3%28An-1-An-2%29%2C%E6%B1%82%7BAn%7D%E7%9A%84%E9%80%9A%E9%A1%B9%E5%85%AC%E5%BC%8F%2C%E5%89%8Dn%E9%A1%B9%E5%92%8CSn%E6%98%AF%E5%A4%9A%E5%B0%91)
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数列{An},A1=4/3,A2=13/9,当n>=3时,An-An-1=1/3(An-1-An-2),求{An}的通项公式,前n项和Sn是多少
数列{An},A1=4/3,A2=13/9,当n>=3时,An-An-1=1/3(An-1-An-2),求{An}的通项公式,前n项和Sn是多少
数列{An},A1=4/3,A2=13/9,当n>=3时,An-An-1=1/3(An-1-An-2),求{An}的通项公式,前n项和Sn是多少
设Bn=An-A(n-1);则B(n-1)=A(n-1)-A(n-2);
An-An-1=1/3(An-1-An-2)
那么Bn=1/3B(n-1);
Bn是公比为1/3的等比数列;
Bn=(1/3)^(n-3)B3;
B3=A3-A2=1/3(A2-A1)=1/3(13/9-4/3)=1/27;
Bn=(1/3)^(n-3)(1/27)=(1/3)^n;
An-A(n-1)=(1/3)^n;
A(n-1)-A(n-2)=(1/3)^(n-1);
...
A2-A1=1/3;
相加得:
An-A1=1/3+...+(1/3)^n=1/3[(1/3)^n-1]
An=(1/3)^(n+1)+1;
Sn=(1/3)^2+(1/3)^3+...+(1/3)^(n+1)+n;
Sn=(1/3)^2[(1/3)^n-1]+n=(1/3)^(n+2)+n-1/9
令Bn=An-An-1,所以BN=1/3BN-1,