设数列an的首项A1=A,A1不等于1/4,且An+1=（1/2）*An,n为偶数An+1=An+1/4,n为奇数,记Bn=A2n-1-1/4,n=1,2,3,4……

Question

设数列an的首项A1=A,
A1不等于1/4,且An+1=（1/2）*An,n为偶数 An+1=An+1/4,n为奇数,记Bn=A2n-1 -1/4, n=1,2,3,4……

Answer 1

(1)A2=1/2A1=A/2
A3=A2+1/4=(2A+1)/4
(2)B1=A1-1/4=A-1/4≠0
Bn=A2n-1 - 1/4
Bn+1=A2(n+1)-1 - 1/4=A2n+1 - 1/4
A2n+1=1/2A2n
A2n=A2n-1+1=A2n-1 + 1/4
∴ A2n+1 = 1/2(A2n-1 + 1/4)
∴ Bn+1 =1/2(A2n-1 - 1/4)
而 Bn = A2n-1 - 1/4
∴ Bn+1 / Bn =1/2
而 B1≠0
∴Bn为等比数列
2(1)b1=a2-a1=5/3-2=-1/3
bn=an+1 -an
bn+1 = an+2 -an+1
∵an+2 =5/3an+1 -2/3an
∴bn+1=5/3an+1 -2/3an -an+1
=2/3(an+1 -an)
∴bn+1 /bn =2/3
∴bn为b1=-1/3,等比为2／3的等比数列
∴bn=-1/3(2/3)^(n-1)