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Sounds like the dispaly, cable or cable connection.
Rule out display issues: If your Mac won't turn on - Apple Support
In or out of warranty you can get a free over the counter 'Apple Service Diagnostics' test and ask specifically for the VST (Video Systems Test)
Genius reservation http://www.apple.com/retail/geniusbar/
check warranty https://checkcoverage.apple.com/
Question: Mac screen becomes blank when i start up