MATHEMATICS AND ORIGAMI

Jesús de la Peña Hernández

7.15 PROGRESSIONS

7.15.1 ARITHMETIC PROGRESSIONS

They are formed by a succession of quantities such that any of them is equal to the im-

mediate preceding one plus another constant quantity called ratio d. The progression is an in-

creasing one if d > 0 and is decreasing if d < 0.Let ́s see an example of the former type.

To build it up by folding we begin with a paper strip of width d and adequate length (see first

picture of Fig.1). From its left end we take the progression ́s first term a1. The obtention of a

square (side d) shown in the 3rd picture for the first time, is the key to get successive terms a 2 ,

a 3 and a 4. It is obvious that the only limit to the number of terms is the strip length.

Last picture of Fig 1 shows how the terms of the progression do grow: it looks like a

flattened bellows. Just by counting and looking at that picture, the most important properties of

arithmetic progressions can be checked.

Last terms ́ value:

an=a 1 +()n− 1 d ; a 4 =a 1 + 3 d

Continuous equidistance between three consecutive terms:

ai+ 1 −ai=ai−ai− 1 ; a 4 −a 3 =a 3 −a 2

One term as the arithmetic media of its preceding and following terms:

a 1

d

a 1 d

a 2

d

a 3

d

a 4

1

a 1

a 2

a 3