Question

a1=3,an+1=an+2,求an

Answer 1

你这题若是a(n+1)=3a(n)/a(n+2)，则没法求解
若是a(n+1)=3a(n)/[a(n)+2]，则可以求解
由
a(n+1)=3a(n)/[a(n)+2]
可得：
1/a(n+1)
=
[a(n)+2]/[3a(n)]
=
1/3
+
(2/3)[1/a(n)]
即：1/a(n+1)
-
1
=
1/3
+
(2/3)[1/a(n)]
-
1
=
(2/3)[1/a(n)]
-
缉范光既叱焕癸唯含沥2/3
=
(2/3)[1/a(n)
-
1]
数列
{
1
-
1/a(n)
}
满足：[1
-
1/a(n+1)]
/
[1
-
1/a(n)]
=
2/3
1
-
1/a(n)
=
[1
-
1/a(1)](2/3)^(n-1)
由于：1
-
1/a(1)
=
1-1/3=2/3
所以：1
-
1/a(n)
=
(2/3)^n
1/a(n)
=
1-(2/3)^n
所以：a(n)
=
1/[1-(2/3)^n]
=
(3^n)/(3^n-2^n)