In a book I'm reading, they've defined a variable $r_i$ like this,
$$ \begin{array}{rcll} r_0 &=&q_1r_1+r_2, & 0 < r_2 < r_1 \\ r_1 &=&q_2r_2+r_3, & 0 < r_3 < r_2 \\ \vdots & \vdots & \vdots & \vdots \\ r_{m-2} &=&q_{m-1} r_{m-1}+r_m, & 0 < r_m < r_{m-1} \\ r_{m-1} &=&q_m r_m. \end{array} $$
Afterwards they use it inverted in a proof like $$ r_i = r_{i-2} - q_{i-1} r_{i-1} $$
I'm trying to show that, $$ r_i = q_{i+1} r_{i+1} + r_{i+2} = r_{i-2} - q_{i-1} r_{i-1} $$ in order to understand what's going on.
But I'm invariably stuck, coming up with all kinds of assumptions and results. One example is,
Assumption: (based on the book) \begin{align*} r_{i-1} &= q_{i} r_{i}, \\ r_{i-2} &= q_{i-1} r_{i-1} + r_{i}, \\ % r_{i+1} &= r_{i-2+1} - q_{i-1+1} r_{i-1+1} = r_{i-1} - q_{i} r_{i}, \\ r_{i+2} &= r_{i-2+2} - q_{i-1+2} r_{i-1+2} = r_{i} - q_{i+1} r_{i+1},\\ \end{align*}
\begin{align*} 0 =& q_{i+1} r_{i+1} + r_{i+2} - r_{i-2} - q_{i-1} r_{i-1}\\ % =& q_{i+1} r_{i+1} + (r_{i} - q_{i+1} r_{i+1}) - (q_{i-1} r_{i-1} + r_{i}) - q_{i-1} r_{i-1}\\ % =& q_{i+1} r_{i+1} + r_{i} - q_{i+1} r_{i+1} - q_{i-1} r_{i-1} - r_{i} - q_{i-1} r_{i-1}\\ % =& q_{i+1} r_{i+1} - q_{i+1} r_{i+1} - q_{i-1} r_{i-1} - q_{i-1} r_{i-1}\\ % % =& 2 q_{i-1} r_{i-1} \\ % =& 2 q_{i-1} ( q_{i} r_{i} ) \\ % =& 2 q_{i-1} q_{i} ( q_{i+1} r_{i+1} + r_{i+2}) \\ % % =& 2 q_{i-1} ( q_{i} r_{i} + r_{i-1} - q_{i} r_{i} ) \\ % =& 2 q_{i-1} r_{i-1} \\ \end{align*}
I'm probably missing something incredibly basic, but I've been at it for two hours, so I've tried. I'm not even sure I'm making the right assumptions anymore, given that the above yields something like $$r_{i+1} = r_{i-1} - q_{i} r_{i} = q_{i} r_{i} - q_{i} r_{i} = 0$$ So any help is appreciated.
[edit] Another attempt, based on the assumption: $ r_{i-1} = q_{i} r_{i} + r_{i+1} $
\begin{align} 0 =& q_{i+1} r_{i+1} + r_{i+2} - r_{i-2} - q_{i-1} r_{i-1}\\ % =& q_{i+1} (r_{i-1} - q_{i} r_{i}) + (r_{i} - q_{i+1} r_{i+1}) - (q_{i-1} r_{i-1} + r_{i}) - q_{i-1} (q_{i} r_{i} + r_{i+1})\\ % =& q_{i+1} r_{i-1} - q_{i+1} q_{i} r_{i} + r_{i} - q_{i+1} r_{i+1} - q_{i-1} r_{i-1} - r_{i} - q_{i-1} q_{i} r_{i} - q_{i-1} r_{i+1}\\ % =& q_{i+1} r_{i-1} - q_{i+1} q_{i} r_{i} - q_{i+1} r_{i+1} - q_{i-1} r_{i-1} - q_{i-1} q_{i} r_{i} - q_{i-1} r_{i+1} - r_{i} + r_{i}\\ % =& q_{i+1} r_{i-1} - q_{i+1} q_{i} r_{i} - q_{i+1} r_{i+1} - q_{i-1} r_{i-1} - q_{i-1} q_{i} r_{i} - q_{i-1} r_{i+1}\\ % =& q_{i+1} (r_{i-1} - q_{i} r_{i} - r_{i+1}) - q_{i-1} (r_{i-1} + q_{i} r_{i} + r_{i+1})\\ % % =& q_{i+1} (r_{i+1} - r_{i+1}) - q_{i-1} (r_{i-1} + r_{i-1})\\ % =& - q_{i-1} (r_{i-1} + r_{i-1})\\ % % &\\ &\\ \end{align}
I just don't seem to get anywhere.
[Edit 2, solution] Move $- q_{i-1} r_{i-1}$ over, increment subscripts twice. \begin{align*} r_i &= r_{i-2} - q_{i-1} r_{i-1} \Rightarrow\\ % r_{i-2} &= r_{i} + q_{i-1} r_{i-1} \Rightarrow\\ % r_{i-1} &= r_{i + 1} + q_{i} r_{i} \Rightarrow\\ % r_{i} &= r_{i + 2} + q_{i+1} r_{i+1}. \end{align*}