Question

关于一道数列问题

某数列
（An）满足2An+1=An+3=0,A1=1.求An.

Answer 1

2a(n+1)=an+3
a(n+1)=an/2+3/2
a(n+1)-3=(an-3)/2
[a(n+1)-3]/(an-3)=1/2
所以{an-3}是以a1-3=-2为首相q=1/2为公比的等比数列
an-3=(-2)*(1/2)^(n-1)=-2^(2-n)
an=3-2^(2-n)

Answer 2

2An+1=An+3推出An+1=An/2+3/2
求出不动点x=x/2+3/2,x=3
两边同减去3得：An+1-3=An/2+3/2-3=1/2(An-3)
设Bn=(An-3),则Bn+1=(1/2)Bn,B1=A1-3=-2,
Bn=(-2)(1/2)^n-1
An=(-2)(1/2)^n-1 +3