APPENDIX I

Pascal’s Trjangle and the Binomial
Theorem

The following triangle shows at a glance the coefficients of
the expansion of any binomial of the form (x+41),

o d R e R ey

These coefficients are derived from the general formula for
the expansion of

n(n
(x—f—a)"=x"+nx”—1a+(I\2)x”“2a2+ L R A

If a=1, then all powers of ¢ are also equal to 1. Any number
multiplied by 1 remains unchanged, so that a and its powers
may be eliminated from the expansion. Then

(x—]~I)":x”+nx"‘1+n—(fif‘21)x””2+ s e

and the coefficients of this expansion are, in order:

TN m VLA g
I,2
it being remembered that 7 represents the value of the power
to which the binomial is being raised. Thus, for different
powers, the coefficients are:
11 161

o
e
B
&
3

2
F

2
R
i
T
¥

:
’..

1