e

&

SERIES—SHAPES AND SQUARES 19

o
N -
PN -
QUL WO N
N O W N M
AR AR

oA

 

H

Totals: 178 "6 31078 "1 128

§EE
ths

T
"

T B L e R R O O R At AR

B
Lt

(i.e. the seventh triangular number is the total of seven
terms).

This knowledge makes it possible to calculate any specified
triangular number without having to calculate all the inter-
mediate numbers. This is done by using the known formula
for the calculation of the totals of Arithmetical Progression

S ___(z%—l)n’ where S is the sum of the progression, a is the

first term, / is the last term and 7 is the number of terms.
Where §'is a triangular number, we know that the first term
is always 1 and that the number which is the last term is
always the same as the number of terms.

2
(a+0n (I—Zn)n i n-+n

Spea e e
_‘Tx.k‘:h IRHBOA fi

Thus S S becomes

so that if we know 7, we also know S. In other words, if we
know the position of a specified triangular number we also
know what that number is. Thus, for the seventh triangular

SRR AL
RO U NR R AR

nsinauns
AL

: e gt ‘
number, n=7% and the required number is —2——=28. :

SUTE Ao
RO

Expressed differently, this relationship can be stated as |
revealing the fact that half the sum of any number and itself |
squared is always a triangular number. ‘

Triangular numbers can also be expressed in either of the
following ways:

iy "'W"W'fl‘l I

I 3 6 10 15 21 etc.
(@) 1x1 2x1} 3X2 4%x2% 5x3 6x3}
(0) 2x3 3gx1 4x1} 5%x2 6x2} 7x3

|
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