MATHEMATICS AND ORIGAMI

Mathematics and Origami

2

= i−^1 + i+^1

i

a a

a ; 3 2 4 1 1 1 1 2 3

2

2 4

2

3

2

a d a

a a a d a d a d

a = + =

+

=

+ + +

=

+

=

Relation between term n and one of its precedent p, being i the number of terms in between the

two:

an=ap+d()i+ 1 ; a 4 =a 2 + 2 d

The sum of two equidistant terms from the extremes, is equal to the sum of these extremes:

a 1 +an=a 1 +i+an−i ; a 1 +a 4 =a 2 +a 3 (i = 1)

Sum of all of the terms of an arithmetic progression:

n

a a

S n

2

=^1 + ; S a a 4 2 a 2 a a a 2 ()a 3 d 4 a 6 d

2 1 4 1 1 1 1

=^1 +^4 = + = + + + = +

7.15.2 GEOMETRIC PROGRESSIONS

They are those in which each term is equal to its immediate precedent multiplied by a

constant r called ratio of the progression. Let ́s see first, one of the increasing type (r > 1). Fig.

2 in point 7.14.3 is one example. In it, the first term is a 1 = 1 and the ratio is a > 1.

If we wish that the first term be a 1 ≠ 1 (keeping the same ratio r = a) we would have to

build an orthogonal spiral parallel to the former one beginning with a 1. By so doing we get Fig.

1 of present point 7.15.2. In it, the value of each term is measured from O to the correspondent

ai. Through the similarity of the triangles shown we can also see that:

4 1

5 r

a

a

= ;

1 1

2 r

a

a

=

Here we have some properties that can be observed in Fig. 1:

Last term as a function of the first one and the ratio:

1

1

= n−

an ar ;

4

a 5 =a 4 r=a 3 r×r=a 2 r×r×r=a 1 r×r×r×r=a 1 r

Sum of the n from the first terms of a geometric progression as a function of the first, the last

and the ratio:

1

1

−

−

=

r

ar a
S n ; measuring on the figure, at a graphic scale, we have: (1)

a 5 = 2. 2458 ; a 4 = 2. 0215 ; a 3 = 1. 8196 ; a 2 = 1. 6378 ; a 1 = 1. 4742

r= 1. 0751 ; 01 = 0. 9677

The value of r to be taken to the sum formula is 1.0751 / 0.9677 = 1.111

a

a 4

a 1 1 o

3

1

r

a 2

a 5