Power cut and computer no longer powers up

Blackout occurred the other night shutting the computer down.

I tried restarting it the other day and it initially starts in that a light comes on and the fan whirs. this last for about 2 seconds and then everything dies.

It then remains dead unless I unplug the power cable at the back and reinsert it whereupon it will do the same again.

Is it likely that I need a new power supply. I am pretty certain the restart button is free. I will get rid of any unnecessary dust in the computer later but otherwise is this the likely culprit?