Example 1 Given the sequence $\left\{a_{n}\right\}$ with the first term $a_{1}=1, a_{n}=$ $2 a_{n-1}+1(n>1)$. Then the general term formula of the sequence $\left\{a_{n}\right\}$ is
$$
a_{n}=
$$
Let $x=2x+1$, we get $x=-1$. Then $a_{n}+1=2\left(a_{n-1}+1\right)$.
Therefore, the sequence $\left\{a_{n}+1\right\}$ is a geometric sequence with the first term $a_{1}+1=2$ and common ratio 2.
Hence $a_{n}+1=2 \times 2^{n-1}=2^{n}$, which means $a_{n}=2^{n}-1$.