=
s

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28 THE FASCINATION OF NUMBERS

The first numbers are in fact all of the form [z(n—1)+1]
where the series is 73, and are themselves in series, thus:
I 3 7 13 21
2 4 6 8
2 2 2

It should also be noted that each cubic series consists of the
same number of terms as is represented by its cube root. The
series for 52, for example, consists of 5 terms.

The full relationships between cube numbers and the
series of consecutive odd numbers may be demonstrated as
follows. Each progression of odd numbers may be shown
in a different way. Thus the series (7+4-9-+11), representing
the value of 32 is the same as (7) +(7+2) +(7+4) and this,
in turn, is the same as (3 Xx7)+(2-+4). This is a constant
property, and the series for any cube #® may be shown as:
(n X first number in series)

+(Sum of progression (2, 4, etc.) having n—1 terms)

or (W)n(n—1) +1]+"—[2+(2—1)2)]
=n[n*—n+1]4+(n—1)(1+n—1)

=n®—n+n +n%—n
=n3
Cube numbers can also be built up as follows; the number -

of terms in each series again being the same as the number’s
cube root.

I13=1
23=1+4%
Shemt egdeng

4°*=1+7+19+37
5*=1+7+19+37+61
The numbers used in these series are called hexagonal
numbers because they may be pictorially represented in
hexagonal form, and these themselves are related back to
triangular numbers to the extent that the (n+1)th hex-
agonal number is obtained by adding unity to six times the