Need a generalized way to solve $127=1+2+2^2+...$ How can I solve this equation generally? I can solve it by checking in my calculator. But I don't know any generalized way.
$$127=1+2+2^2+2^3+....+ 2^{x-1}$$

The most intuitive way:
\begin{align*}
&\;\underbrace{1 + 1} +2+2^2+2^3+ \cdots + 2^{x-1} \\
&= \underbrace{2 + 2} + 2^2 + 2^3 + \cdots + 2^{x-1} \\
&\;\;\;= \underbrace{4 + 2^2} + 2^3 + \cdots + 2^{x-1} \\
& \; \\ & \; \\ %vertical space
&\quad \quad\quad \quad\quad \; \; \; \; \; \cdots \\
& \; \\ & \; \\ %vertical space
&\quad \quad \quad \quad = \underbrace{2^{x-1} + 2^{x-1}} \\
&\quad \quad \quad \quad \quad \quad = 2^x \\
\end{align*}
Hence,
$$
1 + 2 + 2^2 + 2^3 + \cdots + 2^{x-1} = 2^x - 1.
$$
