> My intuition was already screaming 'it'll be the other way around for a geometric progression' before I read that far, but I'm damned if I can understand or even speculate why.
It's a question of forcing the number of sums/products to be low. An arithmetic progression forces a large number of identical sums for the obvious reason: (a+b) = ((a-k)+(b+k)) = ((a-2k) + (b+2k)), and so on, and those differences of k are... the definition of an arithmetic progression.
The reason a geometric progression produces a lot of identical products is exactly the same. (ab) = (a/k · bk) = (a/kk · bkk)...
It's a question of forcing the number of sums/products to be low. An arithmetic progression forces a large number of identical sums for the obvious reason: (a+b) = ((a-k)+(b+k)) = ((a-2k) + (b+2k)), and so on, and those differences of k are... the definition of an arithmetic progression.
The reason a geometric progression produces a lot of identical products is exactly the same. (ab) = (a/k · bk) = (a/kk · bkk)...