Chapter 27

Figurate numbers

27.1 Triangular numbers ..................

Thenth triangular number is

Tn=1+2+3+···+n=

1

2

n(n+1).

The first few of these are 1 , 3 , 6 , 10 , 15 , 21 , 28 , 36 , 45 , 55 , ....

27.2 Special triangular numbers ...........

Triangular numbers which are squares

Then-th triangular numberTn=^12 n(n+1)is a square, say,m^2 if and
only if
(2n+1)^2 −2(2m)^2 =1.

(

nk+1

mk+1

)

=

(

34

23

)(

nk

mk

)

+

(

1

1

)

,

(

n 1

m 1

)

=

(

1

1

)

.