A sequence (x_n) is defined recursively as (x_n+1)=(6(x_n)+1)/(2(x_n)+5) with x1=3 for n є N
Prove (x_n) is monotone
By looking at the sequence it is decreasing in a monotone fashion but how do i construct a proof to show this?
We form the difference::
which after doing some calculations it is equal to:
and so we have:
.................................................. .........................................1
Now we see that every term in (1),except ,is +ve because for all, n ,which we can prove easily by induction.
So if we can prove for all n ,then (1) will be -ve and hence the sequence decreasing.
And :
next suppose
SO THE sequence is decreasing