ST R T

AR R e U L

20 THE FASCINATION OF NUMBERS

that is, as the product of two factors which are themselves
members of arithmetical progression. That they can be so

expressed is clearly shown by the equation n(iji) as in (a)

and (n+ 1)5 asin (), since these two equations are obviously

: . (n41)(n
two aspects of the same relationship (J_—Q)()

Triangular numbers are closely related to square numbers
in at least two ways. Every triangular number is of such a
nature that if it is multiplied by 8 and added to 1, the result

is always an odd square number. This can be proved as fol-

2
lows. Each triangular number is of the form %z If this is

multiplied by 8 and added to 1, the resultant number is of the
form 4n%+ 4n 1. But this can be factorized to (2n+1)(2n+1)
or (2n+1)2 and since 2n-+1 must be odd, then the resulting
number must be the square of an odd number.

The sum of any two consecutive triangular numbers is
always a square number, and all square numbers are formed
in this way. This can most easily be represented thus:

I 900N 10 I8 aF i,
ot L e e

R TR
There is a simple proof that this is so. It has been shown
n(n+1) n2+n

that the nth triangular number is T e Similarly

 

V(Efln—l +1
2

the (n—r1)th triangular number is ) or

L
2" Therefore the sum of the nth and (n—1)th triangular

numbers is
n®+n n®—n 2n?

:—=n2

2 2 2