Let's say I wanted to find the word 'banana' in a file, then add a line later, such that there were three lines between the 'banana' line and my new line. I could do this:

#!/usr/bin/env perl
use 5.010; use warnings; use strict;

while(<DATA>){
    print;                               # print the line
    if(/banana/){                        # if it contained banana
        print scalar <DATA> for 1..3;    # print the next three lines
        print "This line was added.\n";  # then add the new line
    }
}                                        # then print the rest of the 
+lines
__DATA__
a is for apple
b is for banana
c is for cantaloupe
d is for dill
e is for elderberry
f is for fennel
g is for grapefruit
h is for honeydew
[download]

You should be able to adapt that for your needs. For production code, you'd want to add something to handle the case where there aren't three lines between 'banana' and the end of the file, but since this looks like a homework assignment, I'll assume that's not necessary. Another improvement would be to make the search word and the number of lines to skip arguments set with variables at the top of the problem or taken from the command line. You could also write it as a subroutine which takes the input filename/filehandle, search word, and number of lines as arguments.