In UNIX, they say that "everything is a file"(stream) -- this means that you can "pipe" the output of one command as the input to another (general format is command1 | command2). You can chain together multiple commands this way (command1 | command2 | command3). It is one of the features which provides the great versatility of UNIX, because there are so many different ways to approach a problem.
The first commenter's answer was the simplest and most efficient (EDIT: as is the third poster's grep -l sequentialInsert /directory/to/search/* but that wasn't yet posted when I started writing this), but since you have to use grep:
You can use the output of ls as the input to a grep command, e.g.
ls | grep sequentialInsert
Or, since you have to use a different directory, you would give the path to that directory, in some form.
if the target directory is in your home directory and was named, for example, "target," I think most login shells have a $HOME variable that holds your home directory, so you could use
ls $HOME/target | grep sequentialInsert
If you are using an environmental variable like $HOME or a variable of your own, I was taught that it is generally safest to use double quotes to ensure proper evaluation although I can't remember the details, but that would be
ls "$HOME/target" | grep sequentialInsert
If you have an absolute pathname for your target directory, you can just use that, e.g.
ls /home/users/user1/target | grep sequentialInsert
One other thing, if you need to recursively check subdirectories contained in that directory, and not just the directory itself, use the recursive option of ls. I think BSD and ATT UNIX are different in this, but in the majority of the UNICES I used, it was just ls -R. Also, you can combine options together (e.g. ls -aR is the same as ls -a -R or ls -R -a or ls -Ra) so if you need to make sure to include all the hidden files (beginning with a dot), you can use both options.
ls -R pathname-to-target-directory | grep sequentialInsert
or
ls -aR pathname-to-target-directory | grep sequentialInsert
You don't need to put the asterisks in sequentialInsert because grep already picks it out of anywhere.
The first commenter also mentioned one of the best resources in UNIX: the man pages. If you are ever in a bind about a command like grep, just type man grep, and the man page will tell you the format and an explanation, and relevant pages.
Another cool thing about the man pages is that you can have the man pages check for a topic for you with "man -k", for example man -k filename and it will list one line about every command it thinks has something to do with filenames. It's only a keyword search, so it's not perfect, but the man pages are one of my favourite things about UNIX because I've learned so much from them that I wouldn't have known.