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. 

Here is what I came upon:
If we have: $a_{n} = \frac{1}{2}(a_{n} + \frac{\lambda^2}{a_{n}})$, then it holds that: $\frac{a_{n} - \lambda}{a_{n} + \lambda} = (\frac{a_{1} - \lambda}{a_{1} + \lambda})^{2^{n-1}}$, solving for $a_{n}$ from the above example we have:
$$\frac{a_{n} - 2}{a_{n} + 2} = (\frac{3 - 2}{3 + 2})^{2^{n-1}}$$
$$a_n - 2 = \frac{a_n + 2}{5^{2^{n-1}}}$$
$$a_n5^{2^{n-1}} - a_n - 2.5^{2^{n-1}} - 2 = 0$$
$$a_n(5^{2^{n-1}} - 1) - 2(5^{2^{n-1}} + 1) = 0$$
$$a_n = 2\frac{5^{2^{n-1}} + 1}{5^{2^{n-1}} - 1}$$
