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Series————Shapes and Squares

Apart from the common simple structure of numbers result-
ing from the formation of groups, there are some special
numbers which can be built up into ‘shapes’, such as
triangular numbers, squares and cubes. These shape num-
bers are formed from specific series of other numbers and
they themselves form other series.

A series is a group of numbers in which each number is
related in a certain and invariable way to the number imme-
diately preceding it and the one immediately following it,
and through these two, to all other numbers in the series.
The two main types of series are called Arithmetical Progres-
sions and Geometrical Progressions.

An Arithmetical Progression is one in which the difference
between any two terms is identical and constant. The num-
bers 2, 4, 6, 8, —, for example, form such a progression be-
cause their common difference between any two consecutive
numbers has the constant value of 2.

Numbers may also be in series even if they have no con-
stant common difference, provided that their differences
themselves form an Arithmetical Progression. This is usually
shown by writing the numbers in one row, then showing their
differences in the next row so that these can clearly be seen to
form a progression with a common difference, thus:

2 5 11 20 32
3 6 9 12
3 3 3

A Geometrical Progression differs from an arithmetical
progression in that each of its terms is related to the term
2 17

 

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