The house consists of n rooms (n>4) located at the vertices of a regular polygon with n sides. Adjacent vertexes are connected by corridors. The first vertex contains the output. The person who was in K's room panics and runs through the corridors in search of an exit. In 1 minute, it passes through one corridor. Selects the direction randomly with probability $\frac{1}{2}$ in one direction or in the other Find the mathematical expectation of the exit time. This formula seems to be valid: $\mathbb{E}[F_k] = \frac{\mathbb{E}[F_{k-1}]+\mathbb{E}[F_{k+1}]}{2} + 1, \mathbb{E}[F_{n+1}] = 0, \mathbb{E}[F_{1}] = 0$
Is this a true fact? How can you strictly argue that this is the case