# How to solve this functional (recurrence) equation ?

1. ### jk22

219
I'm in a problem where I have to solve the following functional equation :

$$F(n)^2=n+F(n+1)$$

Does anyone know some methods to solve this kind of problems ?

A similar equation happens in Ramanujan example of root denesting : http://en.wikipedia.org/wiki/Nested_radical#Square_roots

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3. ### mathman

6,387
Don't know of a method. As stated the problem is incomplete - you need an initial condition (F(0) = ?).

4. ### jk22

219
The problem is to find F(1), knowing that $$F(1)=\sqrt{1+\sqrt{2+\sqrt{3+\ldots}}}$$.

5. ### Mentallic

3,657
Should that be to find F(n) given that $F(1)=\sqrt{1+\sqrt{2+\sqrt{3+\ldots}}}$ ?

Find F(2) and the pattern becomes clear.

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