Question

数列{an}满足a1=1，a2=3/2，an+2=3/2an+1-1/2an

Answer 1

a(n+2)=(3/2)a(n+1)-(1/2)an
a(n+2)-(1/2)a(n+1) = a(n+1) - (1/2)an
= a2-(1/2)a1
= 1
a(n+1) - (1/2)an = 1
a(n+1) -2 = (1/2)(an -2)
(an -2) = (1/2)^(n-2). (a2-2)
= -(1/2)^(n-1)
an = 2- (1/2)^(n-1)

Answer 2

an+2 - an+1=1/2an+1 - 1/2an
(an+2 - an+1)(an+1 - an)=1/2
数列｛an - an-1｝为公比是1/2的等比数列
an - an-1=(a2-a1)1/2^(n-2)=1/2^(n-1)
所以得
an-1 - an-2=1/2^(n-2)
....................
a3-a2=1/2^2
a2-a1=1/2 左边和左边相加，右边和右边相加，得
an-a1=1/2^(n-1)+1/2(n-2)+.....+1/2^2+1/2
an=1/2^(n-1)+1/2(n-2)+.....+1/2^2+1/2+1/2^0
an=(1-1/2^n)/(1-1/2)
=2-1/2(n-1)
当n=1时a1=2-1=1
当n=2时a2=2-1/2=3/2
所以an=2-1/2(n-1) （n=>1，n属于正整数）