Question

数列{xn}满足xn+1=1+2/xn(n∈R),x1=11/7

求证:数列{1/(xn-2)+1/3}是等比数列，并求出xn的表达式

Answer 1

记{an}={1/(xn-2)+1/3}
通分化简:an
=
1/(xn
-
2)
+
1/3
=
(xn
+
1)/3(xn
-
2)
将
xn+1
=
1
+
2/xn
代入
an+1
=
1/(xn+1
-
2)
+
1/3,:
an+1
=
1/(xn+1
-
2)
+
1/3
=
1/[1
+
2/xn
-
2]
+
1/3
通分化简得:an+1
=
2(xn
+
1)/3(2
-
xn)
an+1/an
=
-2
因此{an}为等比数列，公比q
=
-2
首项
a1
=
1/(x1
-
2)
+
1/3
=
-2
{an}的通项公式:an
=
a1*q^(n-1)=
(-2)^n
an=1/(xn
-
2)
+
1/3
=
(-2)^n
解得
xn
=
1/[(-2)^n
-
1/3]
+
2