I am supposed to prove that if $n$ players play in $n-1$ ways. Clearly the base case is trivially true, whence only $1$ match will be played. Also the case when the number of players is of the form $2^k$ where $k\in \mathbb{Z}^{+}$, can be easily formulated as follows. $$2^{k-1}+2^{k-2}+2^{k-3}+\ldots+2+1=2^{k}-1$$

I am not sure how to use induction on it in the general case, including the case when $n$ is odd. Any hints are appreciated. Thanks.