I have a folder with files named as file_1.ext...file_90.ext. I can list a range of them with the following command:
file_1.ext...file_90.ext
$ ls /home/rasoul/myfolder/
Brace expansion happens before variable expansion.
(Moreover, don't parse ls output.). You could instead say:
for f in $(seq $st $ed); do echo "${DIR}/file_${f}.ext"; done