Arithmetic Series

An arithmetic series is the sum of a sequence , , 2, ..., in which each term is computed from the previous one by adding (or subtracting) a constant . Therefore, for ,

(1)

The sum of the sequence of the first terms is then given by

			(2)
			(3)
			(4)
			(5)
			(6)

Using the sum identity

(7)

then gives

(8)

Note, however, that

(9)

so

(10)

or times the arithmetic mean of the first and last terms! This is the trick Gauss used as a schoolboy to solve the problem of summing the integers from 1 to 100 given as busy-work by his teacher. While his classmates toiled away doing the addition longhand, Gauss wrote a single number, the correct answer

(11)

on his slate (Burton 1989, pp. 80-81; Hoffman 1998, p. 207). When the answers were examined, Gauss's proved to be the only correct one.

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