## (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