0t s 3 R G P 5 b s
} ¢ htikieedis LR L TR L u BRSIE T I AR R SRR Y

PRI R TR e

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4
2

102 THE FASCINATION OF NUMBERS

The cycle may therefore be represented as the sum of an
infinite geometric progression whose first term is 7 and whose
common ratio is 50. All recurring decimal cycles can be built
up in the same way. Their first terms and common ratios are
found thus. Each cycle is equivalent to a vulgar fraction (in
the above case, 1). Convert the fraction into another in
which the last digit of the divisor is g (thus 1 becomes %)
The new numerator (7) is then the first term of the geometric
progression, and the common ratio is the new divisor added
to unity (49+1=50).

Investigation of other cycles is most illuminating. A few
examples follow. The results of various stages of multiplica-
tion are now given in such a way as to render comparison of
digits easier.

Deriwative of - = -0588235294117647 (Digital
root=g)
Multiply by 2 =-1176470588235294
» 5 3 ='1764705882352041
» o 4= '2352941176470588
2 o» 5= *2041176470588235

Dervative of 5 =-052631578947368421  (Digital
root =g)
Multiply by 2 =-105263157894736842
SRR '157894736842105263

In all the foregoing examples (1, &, % and ;) the deriva-
tives have another feature in common. They all consist of an
even number of digits. If these are separated into two groups
by a line drawn between the two central digits, then each
digit in the first group when added to the digit in the same
position in the second group will give the number g.

142857 142-857=999
076923 076-+923=999