Here's how I would look at it:
F=ma tells us that the resultant force F equals ma. We know the mass m - it is the sum of the lift (500 kg), the passengers (560 kg) and the cable (420 kg), which comes to 1480 kg. We know a, because we're told that it is 1.8 m/s^2.
So what is F? It is the resultant of the upwards force T, due to the tension in the cable, and the downwards weight W of the lift, passengers and cable, which is 1480*g N. In other words, F = T-W. Putting this together, we get
F = ma -> T-W = ma -> T = W + ma -> T = 1480*g + 1480*1.8 -> T = 17168 N,
which is the answer you want once you've taken significant figures into account.
This isn't really any different from the answer given above, but the point that makes it confusing is that the question talks about calculating the maximum tension in the cable. The thing to notice is that once the lift starts ascending, some of the cable is above the point that we are considering, and so the weight of that portion of the cable no longer counts in our calculation. This will give a smaller value for the tension. Therefore, we get the maximum possible value for the tension at the point where the lift first starts accelerating, when we can still include the entirety of the cable in our calculations.