Distinct-positive-integer subset-sum lower bound ($|\{{\sum }_{i\in A}{s}_i:A\subseteq [n]\}|\ge 1+n(n+1)/2$)

For $0<s_1<\cdots <s_n$, adding the largest integer creates at least $n$ new sums of subsets. If $P={\sum }_{i<n}{s}_i$ is the previous maximum, the values $P+s_n$ and $P+s_n-s_i$ for $i<n$ are distinct new sums above $P$. Induction gives the displayed bound. Equality holds for $s_i=i$, whose subset sums fill the interval of integers from zero to $n(n+1)/2$.