My laptop aspire 7520 when powered on the fan spins for 5sec and turns itself off

Take out the battery then press and hold power button for about 30 seconds, then put battery back in again and try to turn on. Might as well leave it without the battery for some time after holding the power button. If that doesn't help try removing RAM and turning it on without it.
By the way.. any beeps?

over heating clean the fan and make sure you can hear it running

yeah, make sure the CPU is not getting heated, check for any loose connections, check the power supply is working, if this is a new computer and it has warranty, I advise to take it back and have it checked or replaced by a professional.

you could be over heating make sure the fan is running,if its not try cleaning it out using a hoover.

Yes there is. Ill teach you how. Turn off your pc, remove the cables and battery. Then press and hold the power button for 8 seconds. Reinsert the cables and battery and turn on the pc. Check the lights on the keyboard if it still on. If it is not, then the problem might be the internal power supply inserted on your laptop or the motherboard itself which is situated inside of your laptop. If you do not know how to reseat or remove the parts of your laptop, then call ASUS for possible repair of it. Bonafide ASUS technicians will handle your concern...

Try external monitor to check video card.
If that OK then might be video ribbon cable or inverter board - worst case - screen itself.

Have you checked the cooling fans for CPU for
Cleanleness and or running?

One week old? Take it back to where you bought it.

clean the board by crc or c40... or you can use coller fan external to avoid to much heating

Mine does this but what I found was that if you hold the power button down for about 10 seconds when you are turning it on, it works.