21an+ bn = an+1
also bn= 2an+ bn or bn+1 - an+1= an___(1)
or bn- an= an-1 (n≥2)________(2)
From (1) and (2) an+2= 2an+1 + an (n≥1)
Again, an+1= an+ bn and (bn+1- bn)/2= an __(3)
which gives 2an+1= bn+ bn+1____(4)
We can also write (3) as (bn+2 - bn+1)/2= an+1 , Combining this with (4) gives
bn+2= 2bn+1+ bn (n≥1)
Now, From the Bold equations it is possible to find an and bn in a closed form.
This kind of equations are known as difference equations.(If you don't know how to solve them try using google)
You will get
an= (1+√2)n+ (1-√2)n2√2
and bn= (1+√2)n+ (1-√2)n2
Now you can easily check that limn→∞bnan = √2
and bn2 = 2an2+1 if n is even and bn2 = 2an2-1 is n is odd.
