Assuming you have read some of the issues of how to run 2 routers in the sticky running a router of each type is the best way to solve that issue.
Don't expect it to really be double that is marketing hype that you only get in special situations. It will run faster assuming you have some another device in your house to talk to. It generally makes little difference on internet since that connection is almost always slower than any wireless.
Make sure you use different channels, the radios still will interfere even if they are running different protocols
I have a new wireless N router, which throttles back the speed, in order to broadcast B and G. If I set it to 'full N' I am supposedly able to double it's speed. The problem is the gamer in the house has a 'G' only Xbox.
If I set the old linksys 'G' router as a connection to the Netgear N router, would that allow me to max the speed on the N and still have a G 'net' in the house?