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pick any real number $x$ and integer $k$ and do the following recursive :

1) $x_0 =x $

2) $x_{n+1} = x_n + \sqrt x_n$

using only $x$ and $k$ how to find the value of $x_k$ without going through recursive computation (meaning a formula) ?!

also i didn't know what tags to attach to this problem so i put just GM ^_^

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    $\begingroup$ Why do you think an explicit, closed-form formula can be given? What is the source of this problem? $\endgroup$
    – Seva
    Commented Jan 2, 2017 at 17:08
  • $\begingroup$ Do you want asymtotics or an algorithm? $\endgroup$
    – Pat Devlin
    Commented Jan 2, 2017 at 22:32
  • $\begingroup$ Mayeb you w'd like to connect this problem to number theory under this question :When is $x_n$ is perfect square , equivalent to :$({x_{n+1}-x_{n}})²=x_{n }$, in any way i don't think if your difference equation converges $\endgroup$
    – zeraoulia rafik
    Commented Jan 2, 2017 at 23:16
  • $\begingroup$ Since your function $f(x)=x+\sqrt{x}$ satisfies f(x)>x for x>0, the sequence diverges for any positive starting value ... $\endgroup$
    – Lasse Rempe
    Commented Jan 3, 2017 at 11:46

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