锂离子电池湿法隔膜:数列{an}满足a1=1 a n+1=1/2an+1/2^n,求通项 an

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a(n+1)=1/2an+1/2^n
an=1/2a(n-1)+1/2^(n-1)
=1/2[1/2a(n-2)+1/2^(n-2)]+1/2^(n-1)
=1/4a(n-2)+2*1/2^(n-1)
.....................
=1/2^(n-1)*a1+(n-1)*1/2^(n-1)
=n/2^(n-1)

an=(2n)/(2^n)

解:∵a(n+1)=(1/2)an+(1/2)^n
∴a(n+1)-(1/2)an=(1/2)^n
(1/2)an-(1/4)a(n-1)=(1/2)^(n+1) (1)
(1/4)a(n-1)-(1/8)a(n-2)=(1/2)^(n+2) (2)
......
[(1/2)^(n-1)]a2- [(1/2)^n]a1=(1/2)^(n+n) (n)
∴(1)+(2)+(3)+...+(n)得:
(1/2)an-[(1/2)^n]a1=(1/2)^(n+1)+(1/2)^(n+2)+...+(1/2)^(n+n)
=[(1/2)^(n+1)][1-(1/2)^n]/[1-(1/2)]=(1/2)^n-(1/2)^(2n)
∵a1=1
∴(1/2)an=1-(1/2)^n+(1/2)^n-(1/2)^(2n)
=1-(1/2)^(2n)
∴an=2-2^(1-2n)