(1)

The table below shows
all the whole numbers
as the base numbers &
the respective triangular numbers
generated from these.

Triangular Numbers Table
Base
number
Triangular
number

•••••



••
•••
••••
•••••

0

(0 x 1)/2 = 0

1

(1 x2 )/2 = 1

2

(2 x 3)/2 = 3

3

(3 x 4)/2 = 6

4

(4 x 5)/2 = 10

5

(5 x 6)/2 = 15

(2)

The tabulation below shows the
addition of each pair of
adjacent numbers per (1),
resulting in all whole numbers
to the 2nd. power.

(0 x 1)/2 + (1 x 2)/2   = 0 + 1 = 1 = 12

(1 x 2)/2 + (2 x 3)/2   = 1 + 3 = 4 = 22

(2 x 3)/2 + (3 x 4)/2   = 3 + 6 = 9 = 32

(3 x 4)/2 + (4 x 5)/2   = 6 + 10 = 16 =   42

(4 x 5)/2 + (5 x 6)/2   = 10 + 15 = 25 =   52





From the tabulation above
one can deduce that in general
in to the 2nd. power is
the sum of 2 triangular numbers
as shown below:

n2  = 

(n – 1)n

2

+

(n – 1)n

2