  For Nakayama algebras $A$, we prove that in case $Ext_A^1(M,M) \neq 0$ for an
indecomposable $A$-module $M$, we have that the projective dimension of $M$ is
infinite. As an application we give a new proof of a classical result from
\cite{Gus} on bounds of the Loewy length for Nakayama algebras with finite
global dimension. For Brauer tree algebras $A$ with an indecomposable module
$M$, we prove that $Ext_A^1(M,M) \neq 0$ implies $Ext_A^i(M,M) \neq 0$ for all
$i>0$.
