# Solution #58: Sum to 30

The expressions 4 + 5 + 6 + 7 + 8, 6 + 7 + 8 + 9, 9 + 10 + 11, (-3) + (-2) + (-1) + ... + 6 + 7 + 8, (-5) + (-4) + (-3) + ... + 7 + 8 + 9, and (-8) + (-7) + (-6) + ... + 9 + 10 + 11 are possibilities.

Three of the foregoing six sums consist of only positive integers. Any sum of consecutive positive integers (a+1) + (a+2) + ... + (a+n) is unchanged by placing the sum (-a) + (-a+1) + ... + (a-1) + a in front of it. Therefore, the three "positive only" sums ensure that three more possibilities are feasible.

Consider the factors of 30: 1 x 30, 2 x 15, 3 x 10, and 5 x 6. All odd factors greater than 1 produce consecutive integral sums, as follows. Take 3 x 10, for example. Three consecutive integers with 10 in the middle will add to 30, which produces 9 + 10 + 11.

Observe that 30 = 0.5 x 60. The consecutive integers 0 and 1 average 0.5. Another sum can be formed by taking thirty pairs of integers each totaling 1. The sum is given by (-29) + (-28) + (-27) + ... + 28 + 29 + 30. The result is actually the sum based on putting integers in front of 30, which comes from 30 x 1. Note that 1.5 x 40, 2.5 x 24, and 7.5 x 8 correspond to some of the other foregoing sums.

Source: Mathematics Teacher