insert/append line after x lines after match

insert line after match

sed '/^root/a new line' /etc/passwd

insert line after 2 lines after match

sed '/^root/!{ p; d }; n;n; i new line' /etc/passwd

insert line after 3 lines after match

sed '/^root/!{ p; d }; n;n;n; i new line' /etc/passwd