Yes you do. According to the specs your laptop supports only B and G. You will have to purchase a USB Dongle Wireless N Receiver to take advantage of N speeds. Try and buy one that is the same brand as the router if possible.
Unless your Dell's wireless adapter is "n", it will only work at "g" speeds with your new router. Keep in mind that your limiting network performance with the Internet is due to the speed of you internet connection. That being the case, there will be no difference in how well your Dell does cruising the 'Net with the "g" or the "n".
On your internal network it will make a difference, just not with the external network.