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26 THE FASCINATION OF NUMBERS

The totals of even series can thus be expressed in three
different ways, as follows:

Series No.of n*4n  (nt+1)2—(n+1) a(nt1) Total of
terms n Series
2 1 1241 22 —2 IX2 2
244 2 28 o0 32—3g 2X3 6
2+4+6 3 3%+3 42 —4 3 X4 12
2+4+6+48 4 4%*+4 52—5 4X%X5 20

The numbers appearing in the total column are called
oblong numbers and can be shown in a pyramid form:

2 6 12 20 30
4 6 8 10
2 2 2

It will be noticed that the oblong numbers are exactly twice
as great as the relative triangular numbers.

Square numbers can also be related to each other in that
the sums of the squares of (n+1) consecutive integers—of
which the greatest is 2z(n+1)—is equal to the sum of the
squares of the next n integers. Thus, if n=2; then 2n(n-+1)
=12; and (n+41) =3, giving:

102+1124-12%2=1324} 142

Similarly, where n=g, we have:

212+4-2224-23%24-24%=2524262{ 242