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18 . THE FASCINATION OF NUMBERS

immediately next to it by a fixed ratio instead of by a com-
mon difference. For instance, the numbers 3, g, 27, 81 ...
form a Geometrical Progression, each term being three times
as great as the preceding term.

It is possible to generate a series simply by addition or
multiplication in conformity with the above principles. The
series which are so closely related to the shape numbers may,
however, be said to exist in their own right, giving valuable
clues to the structure of various types of numbers.

The first such series of importance is that of the triangular
numbers which form the progression 1, 3, 6, 10, 15, ...
Why these are called triangular numbers is more easily
understood if they are represented pictorially thus:

I 3 6 15

It will be seen that each number is made up from the pre-
vious number by adding a further row at the bottom of the
design, and that this row of necessity increases by one at each
move in order to support the previous number. It follows
from this fact that triangular numbers are made up thus:

I=1
3=I-+2
6=1+42+3

10=1+2+3+4

and that these are obviously all of a pattern, each being
an arithmetical progression carried one stage further than
the previous number. Triangular numbers can therefore be
represented as addition sums, as on page 19.

From this it is apparent that not only is each triangular
number the sum of an Arithmetical Progression, but also that
the number of terms in each Progression is the same as the
relative position of the triangular number in its own series