Central simple k-algebra: What it says on the tin.
Brauer equivalence: $A\sim B$ if there exists $m,n\in\Bbb Z_{\geq 1}$ such that $A\otimes_k M_m(k)\cong B\otimes_k M_n(k)$.

Azumaya $R$-algebra ($R$ commutative): An $R$-algebra with centre $R$, which is finitely generated and projective as an $R$-module, and such that $A\otimes_R A^{op}\cong End_R(A)$ via the homomorphism $a\otimes b^{op}\mapsto (c\mapsto acb)$.
Brauer equivalence: $A\sim B$ if there exists finitely generated projective faithful $R$-modules $E$ and $E'$ such that $A\otimes_R End_R(E)\cong B\otimes_R End_R(E')$

Any path from $1$ to $n+m$ passes through $n$ or $n+1$

There are $f_{n+1}$ paths from $1$ to $n+1$

Number of paths from $n+1$ to $n+m$ is $f_{(n+m) - (n+1) + 1} = f_{m}$

So there are $f_{n+1} f_{m}$ paths from $1$ to $n+m$ that pass through $n+1$.

There are $f_{n}$ paths from $1$ to $n$.

Number of paths from $n$ to $n+m$ is $f_{(n+m) - n + 1} = f_{m+1}$.

However, $f_{m}$ of these paths go through $n+1$, and hence are already accounted for in the earlier part.

So there are $f_{m+1} - f_{m} = f_{m-1}$ paths from $n$ to $n+m$ that do not pass through $n+1$.