<s>[INST] How does recurrence $a_n=\frac12(a_{n-1}+\frac\beta{a_{n-1}})$ become $b_n=\frac12\frac{b_{n-1}^2+\beta-\alpha^2}{b_{n-1}+\alpha}$ when $b_n=a_n-a$? 
$a_n = \frac{1}{2}(a_{n-1} + \frac{\beta}{a_{n-1}})$ for $ n > 0$ with $a_0 = 1$. Changing variables in this recurrence, and letting $b_n = a_n - a$, we find by simple algebra that $b_n = \frac{1}{2}\frac{b_{n-1} ^2 + \beta - \alpha^2}{b_{n-1} + \alpha}$.

I've tried making the substitutions and doing algebra but can't come up with it. Would somebody please help show how to get from the expression for $a_n$ to the one for $b_n$ using simple algebra?
 [/INST] $b_n = a_n - a$, $b_{n-1} = a_{n-1} -a$. Hence
$$\begin{align}b_n = a_n - a &= \frac12(a_{n-1} + \frac \beta{a_{n-1}})-a
\\&=\frac12(b_{n-1}+a + \frac \beta{b_{n-1}+a})-a
\\&=\frac12(b_{n-1}+a + \frac \beta{b_{n-1}+a}-2a)
\\&=\frac12(b_{n-1}-a + \frac \beta{b_{n-1}+a})
\\&=\frac1{2(b_{n-1}+a)}((b_{n-1}+a)(b_{n-1}-a)+\beta)
\\&=\frac{b_{n-1}^2-a^2+\beta}{2(b_{n-1}+a)}
\\&\left(=\frac12\frac{b_{n-1}^2+\beta-a^2}{b_{n-1}+a}\right)
\end{align}$$
</s>