Given $m = 2n + 1$, what integer between 0 and $m$ is the inverse of 2 modulo $m$? Answer in terms of $n$.

$$\small{\text{ Given $m = 2n + 1$, what integer between 0 and $m$ is the inverse of $ 2 \pmod{ m}$ }}\\ \small{\text{ Answer in terms of $n$. }}$$

$$\small{\text{ $2x \equiv1 \pmod{m}$ }}\\ \small{\text{ $2x \equiv1 \pmod{2n+1}$ }}\\ \small{\text{ $2x-1 = 2n+1$ }}\\ \small{\text{ $2x = 2n+2$ }}\\ \small{\text{ $\mathbf{x = n+1}$ }}\\$$