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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 d. Therefore, for k>1,

a_k=a_(k-1)+d=a_(k-2)+2d=...=a_1+d(k-1).
(1)

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

S_n=sum_(k=1)^(n)a_k
(2)
=sum_(k=1)^(n)[a_1+(k-1)d]
(3)
=na_1+dsum_(k=1)^(n)(k-1)
(4)
=na_1+dsum_(k=2)^(n)(k-1)
(5)
=na_1+dsum_(k=1)^(n-1)k.
(6)

Using the sum identity

sum_(k=1)^nk=1/2n(n+1)
(7)

then gives

S_n=na_1+1/2dn(n-1)=1/2n[2a_1+d(n-1)].
(8)

Note, however, that

a_1+a_n=a_1+[a_1+d(n-1)]=2a_1+d(n-1),
(9)

so

S_n=1/2n(a_1+a_n),
(10)

or n 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

1/2(100)(1+100)=50·101=5050
(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.

See also

Arithmetic Progression, Common Difference, Geometric Series, Harmonic Series, Prime Arithmetic Progression Explore this topic in the MathWorld classroom

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References

Abramowitz, M. and Stegun, I. 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.Burton, D. M. Elementary Number Theory, 4th ed. Boston, MA: Allyn and Bacon, 1989.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.Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. New York: Hyperion, 1998.Pappas, T. The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 164, 1989.

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Arithmetic Series

Cite this as:

Weisstein, Eric W. "Arithmetic Series." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ArithmeticSeries.html

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