Q: why does my computer come on but my screen won't light up

Hard to say. Could be display itself,connecting cable or video card.

It is better to take the computer to Apple Store for evaluation.

Mac is under warranty for 1 year from the day of purchase, if you have AppleCare it is 3 years.

AppleCare: 1-800-275-2273

But you can try Reset SMC.

http://support.apple.com/kb/HT3964

Choose the method for:

"Resetting SMC on portables with a battery you should not remove on your own".

Best.

Check out KB Article MacBook Pro: Distorted video or no video issues to see if your model comp qualifies.