The equation is: nth element = 2 * (n - 1)th element.I know basic sum distinct set is Z =(1 2 4 8 16 32 64 and so on) which follows the equation like : Nth element = (n-1)th element + (n-2) element + 1.

You probably googled for "distinct subset sums." I found these two articles (PDF). Apparently, there is Erdős conjecture that for some constant c, the maximum of an n-element set with distinct subset sums is at least c * 2^n. As you noted, there are n-element sets where the maximum is 0.5 * 2^n. The second article above proves that there are sets where the maximum is < 0.22002 * 2^n for sufficiently large n.