Characteristic equation of a recurrence relation? I am trying to find the general term of the following recurrence relation:
$$a_{n + 1} = \frac{1}{2}(a_{n} + \frac{1}{a_{n}})$$
where $a_1 = 3$. I'm failing to write the characteristic equation. 

This is the recurrence relation of the Heron method for computing the square root of $1$, also seen as the application of Newton's method to $a^2=1$.
As is shown by the relation
$$\frac{a_n-1}{a_n+1}=\frac1{2^{2^{n}}},$$
it has quadratic convergence and virtually equals $1$ for all $n$.

Indeed,
$$\frac{a_{n+1}-1}{a_{n+1}+1}=\frac{\dfrac{a_n^2-2a_n+1}{2a_n}}{\dfrac{a_n^2+2a_n+1}{2a_n}}=\left(\frac{a_n-1}{a_n+1}\right)^2$$ and by induction
$$\frac{a_{n}-1}{a_{n}+1}=\left(\frac{a_0-1}{a_0+1}\right)^{2^n},$$
and 
$$a_n=1+\frac2{2^{2^n}-1}.$$
