Abstract
The product of the first $n$ terms of an arithmetic progression may be developed in a polynomial of $n$ terms. Each one of them presents a coefficient $C_{nk}$ that is independent from the initial term and the common difference of the progression. The most interesting point is that one may construct an "Arithmetic Triangle'', displaying these coefficients, in a similar way one does with Pascal's Triangle. Moreover, some remarkable properties, mainly concerning factorials, characterize the Triangle. Other related `triangles' -- eventually treated as matrices -- also display curious facts, in their linear \emph{modus operandi}, such as successive "descendances''.
Highlights
What is the product of the first n terms of an arithmetic progression? Contrary to geometrical progressions, where simple equations allow to calculate the sum or the product of its first n terms, the issue concerning the product for an arithmetic progression is not so easy
Since I never returned to the subject till recently, when a student asked me if I knew a formula for the product of the first n terms of an arithmetic progression: a1, a2 = a1 + d, a3 = a1 + 2d, . . . , an = a1 + (n − 1) d
= Cn; n−k+1 or and that one may generate a triangle formed by these numbers, using a similar recurrence relation to the one established by theorem 1, which is essentially symmetrical to the Arithmetic Triangle!
Summary
What is the product of the first n terms of an arithmetic progression? Contrary to geometrical progressions, where simple equations allow to calculate the sum or the product of its first n terms, the issue concerning the product for an arithmetic progression is not so easy. Since I never returned to the subject till recently, when a student asked me if I knew a formula for the product of the first n terms of an arithmetic progression: a1 , a2 = a1 + d , a3 = a1 + 2d , . Several papers have been recently produced concerning Stirling numbers and cyclic groups, for instance (Broder, 1984) and (Deveci & Akuzum, 2014); or Pascal matrices (Deveci & Karaduman, 2012) and (Hiller, 2016). They don’t seem especially relevant to this article, which essentially deals with basic stuff
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