Finding the common difference in an arithmetic sequence for a special case

Question

Suppose that $m^2S_m, mS_{m^2}, S_{m^3}$ are three arbitrary terms in an arithmetic sequence. These terms are also three successive terms in an geometric sequence. If in the arithmetic sequence, $S(20) = 20$, how can we find the common difference of this arithmetic sequence? Looking for the answer for a while, this question was originally asked by my friend.