n

 

i

BT UIE N RN RN TR KRS KPR Rt | -
T it BTN RS IR, il

P - "
B e e LT S e Uittt

' . FRE T
R PRTPIERTTE e a1 LRI TREFTTY TR LR " il

k;v"

b AATASOHNE o s

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g

 

162 THE FASCINATION OF NUMBERS
Power Cocfficients

I I I

2 I 2 I

3 I 3 3 I

+ g Bk

and the construction of the triangle becomes apparent.

Once the building of the triangle has been started, how-
ever, there is no need to use the full formula in order to
ascertain the coefficients for any specified expansion. The
required coefficients may be derived from the triangle itself.
The first and last coefficients are both invariably 1. The
other coefficients of any one expansion may be derived from
the expansion one degree lower—merely by adding together
the coefficients of the latter taken in pairs. The coefficients
of the expansion where n=4, are:

LR
Taking these in pairs, we have:

1+4= 5 4+6=10
6+4=10 4+I= 5§
and the coefficients of the expansion where n=5 are there-
fore:
F5 LRl
Another binomial expansion may be employed in a
method for the extraction of the roots of numbers. In the ex-
pansion of (1-+#)" where 7 is integral and positive, all the
terms of the expansion are also positive, but when 7 is frac-
tional and less than unity, the terms of the expansion which
contain powers of x are alternately positive and negative.
This distinction need not concern us here, however. The
expression (I +x)" may be calculated approximately by taking
only the first two terms of its expansion, if x is very small. In
these circumstances, the subsequent terms are too small to
have any significant effect on the result.

Thus: (1 +x)"=1-nx approx.

= Ea5i] SRR TR

REITE it IIE
s LRtk Sh NS