You could just write down his daily earnings:
20
20+d
20+2d
20+3d
20+4d
20+5d
Adding these, we get 120 + 15d. So 120+15d = 1620, and 15d = 1500, so d = 100. With the value of d, you can answer any question - we can see above he won 20+5d on the last day, so he won 520 on the last day.
I prefer a different approach. He won an average of 1620/6 = 270 per day. Our set is equally spaced (increasing by a constant d each day); in an equally spaced set, the average of the smallest and largest things is always the same as the average of the entire set. The smallest value is 20, and the largest is what we want to find. If we call the largest value x, we then know that the average of 20 and x is equal to 270:
(20+x)/2 = 270
20 + x = 540
x = 520.
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