" PO

"

YL eIt

e L Gt T TR T e e

R T RN R et UL AR
P RTEEN

&‘.'
e
&
%
&

 

166 THE FASCINATION OF NUMBERS

where 3, 6 and 10 and all similar numbers are triangular.
If we treat the triangular numbers as terms of series and
find the sums of these series, we have:

1+3+6 =10
1+3+6-+10 =20

1+3+6+10+15=35
From this we derive the new series 1, 4, 10, 20, 35, etc., and
these numbers are found to be equal to the number of com-
binations of things taken three at a time:

e s R M Ty
e Xk 10 0207 188
From these facts it is clear that "C,, for example, may be ex-
pressed as an addition of terms of the form "C,. Thus:

4Cs BE 202 *+ 302
and similarly 7Cy;=2C,+3C,+*C,+°C,+°C,
Numerically, the latter example is expressed as:
SO S R i B
CaaTrat
_(2.1)+(3-2) +(4-3) +(5-4) +(6.5)
2

 

an expression which, despite its obvious pattern, would be
difficult to prove (short of calculating each side in full) with-
out the use of the combination formula allied to the theory
of the nature of triangular numbers.