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AT T A 1T T4 T4 4

AR

ATt v SRR s e LA R
diadida ey AR RN RTITR

22 THE FASCINATION OF NUMBERS

The foregoing relationship can be proved by the formula
for the sum of arithmetical progression =%(a+L)n.

In the present series, a is always 1; L is always 2n—1.

Therefore, sum of series=%(a+L)n

=3(14+2n—1)n
=1(2n)n=n?

The knowledge that all square numbers may be derived by
the progressive addition of all odd numbers from 1 upwards,
leads to a way of finding values for #, y and z in the equation
X2 4p2=22,

Examine first 32442=52
This can be written :
(14+3+5)+(1+3+5+7) =(14+3+5+7+9)

or . (1+3+5)+(1+3+5+7)=(1+3+5+7)+9
The two sides agree because _
(@) (14+3-+5+7) appears on both sides
(6) 1+3+5=9

Thus the right-hand number z2 will always be the sum of
two other squares (+? and %) when the last term in its pro-
gression is itself a square, being indeed the square of x; and
y=z—1.

To take a further example, the next odd square after 9 is
25. Therefore the sum (z2) of the progression of which 2 5 1is
the last term will be the sum of two other squares (x* and %)
one of which (x?) will be 25, whence x=5. 22 will be
#(1+25)13=169=132 and the other square (»?) will be
(13—1)2=122

Thus 5%+122=132

Another aspect of the foregoing is the fact that where the
sum of two consecutive numbers is a square, the difference
between their squares is also a square.

Thus 54+4=9
and 5%t Ry

Square numbers can also be related to certain others by
the Pythagorean formula a?-+5%=¢? where ¢ is the hypo-