Permission denied when trying to append a file to a root owned file with sudo [closed]

Answer 1

You are trying to write the result of the command sudo cat add_file to the file /etc/file. And apparently you don't have that right.

man sudo gives that example :

   To make a usage listing of the directories in the /home partition.  Note
   that this runs the commands in a sub-shell to make the cd and file
   redirection work.

    $ sudo sh -c "cd /home ; du -s * | sort -rn > USAGE"

So you should try :

sudo sh -c "cat add_file >> /etc/file"