If a, b, and c form a geometric progression then b = ar and c = ar^2 (for some ratio r):

has to be a perfect cube, or has to be a perfect cube. There are a few possibilities for a (35, 28, 9), but the only one that will work in our case is if a = 9.

If a = 9 and b = 36, then r = 4, which makes c = 144. So a + b + c = 189.