(1)

The column below shows
all the triangular numbers.

0
1
3
6
10



(2)

The tabulation below shows
the numbers per (1)
multiplied by 2

2 x 0 = 0
2 x 1 = 2
2 x 3 = 6
2 x 6 = 12
2 x 10 =20
5 x 13 = 65
7 x 25 = 175
9 x 41 = 369




(3)

The table below shows
the numbers per (2)
as the base numbers &
the respective triangular numbers
generated from these.

Triangular Numbers Table
Base
number
Triangular
number

•••••



••
•••
••••
•••••

0

(0 x 1)/2 = 0

2

(2 x 3)/2 = 3

6

(6 x 7)/2 = 21

12

(12 x 13)/2 = 78

20

(20 x 21)/2 = 120

(4)

The tabulation below shows
the numbers per (3)
multiplied by 10
& the resulting product
increased by 1.

10(0) + 1 = 1
10(1) + 1 = 31
10(3) + 1 = 211
10(6) + 1 = 781
10(10) + 1 = 2101




(5)

The tabulation below shows
all the sub-totals of
the numbers per (4),
resulting in all whole numbers
to the 5rd. power.

1 = 15
1 + 31 = 32 = 25
1 + 31 + 211 = 243 = 3 5
1 + 31 + 211 + 781 = 1024 = 45
1 + 31 + 211 + 781 + 2101 = 3125= 55




From the tabulation above
one can deduce that in general
n to the 5th. power is
the summation of triangular numbers
per the formula shown below:

n5 =

n-1
Σ
k=0

5[ k(k + 1) ][ k(k + 1) + 1 ] + 1