SERIES—CUBES AND OTHERS 31

(After two steps, the series is reduced to a progression with a
common difference of 6.)

FOURTH POWERS

I 16 81 256 625 1296
15 65 175 369 671
50 110 194 302
60 84 108
24 24

(After three steps, the series is reduced to a progression with
a common difference of 24.)

As the power is raised by 1 degree, the number of steps is
increased by 1 and the resulting common differences are
always related. Thus in the series of square numbers, 1 step
is required and the common difference is 2 (or 2 X1 as it
really is). For cubes, the number of steps is 1 4+1 (=2) and
the common difference is g3 X2 X1 (=6). For fourth powers,
the steps are 14141 (=3) and the common difference is
4 X 32X 3 (=94,

The series for all power numbers are similarly related to
each other. Each successively higher power series requires
one more step than the previous series before it reduces to a
common difference; and the common difference for any one
series is found by multiplying the common difference of the
next lower series by the same number as is represented by the
index of the power number.

This means that in the series of numbers raised to their nth
powers, there will be n—1 steps before a common difference
is revealed, and this common difference will be equivalent to
(n)(n—1)(n—2) ... (2)(1).

The respective common differences, 2, 6, 24, 120, as well
as the common difference just shown for nth power numbers,
themselves form the series of factorial numbers which are of
great importance in the binomial theorem and the theory of
probability. The factorial number 7 is usually written n! and
is equivalent to the product of all the integers from 1 to =
inclusive; so that, for example, 4!=1 X2 X3 X 4.

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