arithmetic series

An arithmetic seriesMathworldPlanetmath is the series, i=1nai, in which each real term has the form ai=ai-1+d for i=2,,n where d is constant. The sum of the sequence is given by the following 12n[2a1+d(n-1)]. In order to find the formula above firstly we express the terms of the sequence, a2,,an in terms of a1 and the constant d. In this case we get a2=a1+d,a3=a2+2d,,an=a1+(n-1)d. Now we express the sum of the sequence by developing the series forward and we have:


Reversely, we develop the series backwards and we get


It is easily seen that by adding the two expressions we get

2Sn=n(a1+an) (1)
Sn=12n(a1+an) (2)

Hence, by substituting an=a1+(n-1)d we get the first formula.

Title arithmetic series
Canonical name ArithmeticSeries
Date of creation 2013-03-22 16:17:58
Last modified on 2013-03-22 16:17:58
Owner georgiosl (7242)
Last modified by georgiosl (7242)
Numerical id 10
Author georgiosl (7242)
Entry type Definition
Classification msc 40A05