4

Series (cont.)—Cubes and Others

It has been shown] that all square numbers may be ob-
tained by the summation of arithmetical progressions of
consecutive odd numbers from the number 1 upwards. Cube
numbers may also be obtained from the same series of con-
secutive numbers, but in a fundamentally different way. If
we write down the series as follows:

14 5% 8 YE¥S IR T Sk ST G
the cube numbers may be found as follows:

13 Take only the first number in the series £
2%  the next two numbers (3+5) =8
33 2 2 » three bb (7+9+II) T 27
4 5 »  nfuwr ,,  (13+15+17+19) = 64
53 2 bb 3 five » (21+23+25+27+29) =125

Thus each cube number is equivalent to the sum of a
series, and each series commences where the previous series
finishes. This is quite different to the procedure for the ex-
traction of square numbers where each successive series
includes all previous series.

It is quite a simple matter to ascertain the first term in each
cubic series. Inspection of the above cube numbers shows that
the first numbers in each series are as follows:

Series First number ~ Derived from

" I (1 X0)+1
28 3 (2x1)+1
3° 7 (3 x2)+1
4® 13 (4 x3)+1
5% 21 (5x4)+1

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