# Thread: Recursion sequence, and its limit

1. ## Recursion sequence, and its limit

Let $a,c>0$, a sequence { $a_n$} is defined:

$a_1=c$

$a_{n+1}=\sqrt{a+a_n}$

(1) For which values of c the sequence gets lower, higher or does not change?
(2) Is there a limit in each case? Find it.

Okay, in the first case it's easy, because then $a_n=c$ for every n, and of course the limit is c.
Only I found some weird expressions for c depending on a:
I found that $a_{n+1}>a_n$ when: $c^2-c, which I don't know how to turn it into a 'nicer' expression of c.

That's before trying to find the limit of them...

Thank you very much!

2. If You suppose that the sequence $a_{n}$ tends to a finite limit, the limit itself is obtained imposing that...

$\lim_{n \rightarrow \infty} a_{n+1} - a_{n}=0$ (1)

... i.e. the solution of the equation...

$x=\sqrt{a+x}$ (2)

... so that is...

$a_{\infty} = \frac{1 + \sqrt{1+4a}}{2}$ (3)

It is important to consider that the limit (3) is independent from $a_{1}=c$ if $c=0$ so that...

a) if $a_{1} then the sequence is monotonically increasing...

b) if $a_{1}=a_{\infty}$ then the sequence is constant...

c) if $a_{1}>a_{\infty}$ then the sequence is monotonically decreasing ...

Kind regards

$\chi$ $\sigma$