Since the meaning of the question can be that xn is a fraction, let xn=an/bn, and an and bn are coprime.

knowable

x 1= 1/2，x2=2/3，x3=3/5，x4=5/8，x5=8/ 13，x6= 13/2 1……

That is, a 1 = 1, a2 = 2, a3 = 3, a4 = 5, a5 = 8, a6 = 13, ...

b 1=2，b2=3，b3=5，b4=8，b5= 13，b6=2 1，…

bn=an+ 1

So xn=an/an+ 1)

An +2 = An+1+ An

In the sequence {x2n}

x2n-x2(n- 1)= a2n/a2n+ 1-a2n-2/a2n =(a2n * a2n-a2n+ 1 * a2n-2)/(a2n+ 1 * a2n)

The denominator is a positive number. For the convenience of writing, I won't count it first, just the numerator.

a2n*a2n-a2n+ 1*a2n-2=(a2n- 1+a2n-2)^2-(a2n-2+2a2n- 1)*(a2n-2)=(a2n- 1)

^2>； 0

therefore

X2n-x2 (n- 1) > 0, and the sequence {x2n} is increasing function.