The goal is to find $$$\sum^{X-1}_{i=0}A^{i}$$$ $$$ mod $$$ $$$m$$$ , and a simple thought is that we can calculate $$$\frac{A^{X}-1}{A-1}$$$. But when we divide $$$A^{X}-1$$$ by $$$A-1$$$ then $$$mod$$$ $$$m$$$, problem occurs: when we calculate $$$A^{X}-1$$$ , it's already $$$mod$$$ $$$m$$$ ,and there's maybe no reverse to it.
To solve this problem , I use a simple method: store $$$A^X$$$ with two numbers : $$$k$$$ and $$$r$$$ ,satisfying
When we calculate $$$A^{x}$$$,we refresh $$$k$$$ and $$$r$$$ by multiplication ,then refresh $$$k$$$ by modulo $$$m$$$ , and refresh $$$r$$$ when $$$r \ge A-1 $$$ : we do $$$k$$$ $$$+=$$$ $$$r/(A-1)$$$ and $$$r$$$ $$$=$$$ $$$r$$$ % $$$(A-1)$$$
