Adding the Terms

Add the 10 terms 2, 5, 8, ..., 29. Pair the first with the last: 2 plus 29 is 31. Pair the second with the second last: 5 plus 26 is 31 too. Every pair makes 31, because one number goes up by 3 exactly as its partner goes down by 3. Ten terms make 5 pairs, so the sum is 5 × 31, which is 155.

Written for any arithmetic sequence of n terms, that is n/2 pairs, each worth the first term plus the last. With an odd n the middle term is half a pair, and the count still works:

S = n/2 × (a(1) + a(n))

The story goes that the young Carl Friedrich Gauss added 1 to 100 in moments this way.