Question

已知数列{an}中,a1=1,an+1=5/2-1/an,求an ,要求不用 bn=1/an-2

Answer 1

a(n+1)=5/2-1/an

= (5an- 2)/(2an)

a(n+1) -2 = (5an- 2)/(2an) - 2

= (an -2)/(2an)

1/[a(n+1) -2] = 2an/(an -2)

= 2 + 4/(an -2)

1/[a(n+1) -2] + 2/3 = 4[ 1/(an -2) + 2/3]

{1/(an -2) + 2/3} 是等比数列,q=4

1/(an -2) + 2/3 = 4^(n-1) .(1/(a1 -2) + 2/3)

= -(1/3).4^(n-1)

1/(an -2) = -2/3 -(1/3).4^(n-1)

an = 2- 1/[2/3 +(1/3).4^(n-1)]