Question

已知Sn=2an-1/2^n求an

已知Sn=2an-1/2^n求an

Answer 1

Sn=2an-1/2^n
n>=2时有:an=Sn-S(n-1)=2an-1/2^n-(2a(n-1)-1/2^(n-1))=2an-1/2^n-2a(n-1)+2/2^n
an=2a(n-1)-1/2^n
an/2^n=a(n-1)/2^(n-1)-1/2^(2n),【二边同除以2^n】
即:
an/2^n-a(n-1)/2^(n-1)=-1/2^(2n)
...
a2/4-a1/2^0=-1/2^2
以上各式相加得：
an/2^n-a1/1=-(1/4^n+...+1/4)=-1/4*(1-1/4^n)/(1-1/4)=1/3*(1/4^n-1)
又a1=S1=2a1-1/2,得a1=1/2
故an/2^n=1/3(1/4^n-1)+1/2
an=1/3*2^n*(1/4^n-1)+2^(n-1)
a1=1/2也符合。