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