Thread: Recursion sequence, and its limit

1. Recursion sequence, and its limit

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

$\displaystyle a_1=c$

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

(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 $\displaystyle a_n=c$ for every n, and of course the limit is c.
Only I found some weird expressions for c depending on a:
I found that $\displaystyle a_{n+1}>a_n$ when: $\displaystyle c^2-c<a$, 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 $\displaystyle a_{n}$ tends to a finite limit, the limit itself is obtained imposing that...

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

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

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

... so that is...

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

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

a) if $\displaystyle a_{1}<a_{\infty}$ then the sequence is monotonically increasing...

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

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

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$