Arithmetic Series

An arithmetic series is the Sum of a Sequence $\{a_k\}$, $k=1$, 2, ..., in which each term is computed from the previous one by adding (or subtracting) a constant. Therefore, for $k>1$,

$$a_k = a_{k-1}+d=a_{k-2}+2d=\ldots=a_1+d(k-1).$$ (1)

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

$$S_n \equiv \sum_{k=1}^n a_k =\sum_{k=1}^n [a_1+(k-1)d]$$

$$= na_1+d\sum_{k=1}^n (k-1) = na_1+d\sum_{k=2}^n (k-1)$$

$$= na_1+d\sum_{k=1}^{n-1} k$$ (2)

Using the Sum identity

$$\sum_{k=1}^n = {\textstyle{1\over 2}}n(n+1)$$ (3)

then gives

$$S_n = na_1+{\textstyle{1\over 2}}d(n-1)= {\textstyle{1\over 2}}n[2a_1+d(n-1)].$$ (4)

Note, however, that

$$a_1+a_n=a_1+[a_1+d(n-1)]=2a_1+d(n-1),$$ (5)

so

$$S_n={\textstyle{1\over 2}}n(a_1+a_n),$$ (6)

or $n$ times the Average 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

$${\textstyle{1\over 2}}(100)(1+100) = 50\cdot 101 = 5050$$ (7)

on his slate. When the answers were examined, Gauss's proved to be the only correct one.

See also Arithmetic Sequence, Geometric Series, Harmonic Series, Prime Arithmetic Progression

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 10, 1972.

Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 8, 1987.

Courant, R. and Robbins, H. ``The Arithmetical Progression.'' §1.2.2 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 12-13, 1996.

Pappas, T. The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 164, 1989.