Sum

A sum combines a finite or infinite sequence of terms by repeated addition. The symbol $\sum$ denotes summation over an index.

For positive weights $w_i$, the weighted mean isIt lies between the smallest and largest $x_i$; the same result holds for integrals with a positive weight function.

The arithmetic mean of $x_1,\dots ,x_n$ isIt is the weighted mean with equal weights.

The harmonic mean of positive numbers $x_1,\dots ,x_n$ is $H=n/({\sum }_{i}1/x_i)$. The Jensen inequality for $1/x$ gives $H\le \overline{x}$, with equality exactly when all entries coincide.

The prefix sums $P_m={\sum }_{i=1}^m{a}_i$ can all be computed by the recurrence $P_m=P_{m-1}+a_m$ in linear time. Any contiguous sum is then $P_b-P_{a-1}$, so it takes constant time after preprocessing.