here is the very pithy sed way of finding every 4th line, as a one-liner that turns 4-line fastq into 2-line fasta:
sed -n '1~4s/^@/>/p;2~4p
http://stackoverflow.com/a/9895102/188963
This is easy to accomplish with awk.
Remove every other line:
awk 'NR % 2 == 0' file > newfile
Remove every 10th line:
awk 'NR % 10 != 0' file > newfile
The NR variable in awk is the line number. Anything outside of { } in awk is a conditional, and the default action is to print.
http://stackoverflow.com/a/9895062/188963
You could possibly do it with sed, e.g.
sed -n -e 'p;N;d;' file # print every other line, starting with line 1
If you have GNU sed it's pretty easy
sed -n -e '0~10p' file # print every 10th line
sed -n -e '1~2p' file # print every other line starting with line 1
sed -n -e '0~2p' file # print every other line starting with line 2