## (1)

The column below shows

all the triangular numbers.

0

1

3

6

10

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## (2)

The tabulation below shows

the numbers per (1)

multiplied by 4

& the resulting product

increased by 1.

4(0) + 1 = 1

4(1) + 1 = 5

4(3) + 1 = 13

4(6) + 1 = 25

4(10) + 1 = 41

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## (3)

The tabulation below shows

the numbers per (2)

multiplied by

respective uneven numbers

starting from 1.

1 x 1 = 1

3 x 5 = 15

5 x 13 = 65

7 x 25 = 175

9 x 41 = 369

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## (4)

The tabulation below shows

all the sub-totals of

the numbers per (3),

resulting in all whole numbers

to the 4th. power.

1 = 14

1 + 15 = 16 = 24

1 + 15 + 65 = 81 = 34

1 + 15 + 65 + 175 = 256 = 44

1 + 15 + 65 + 175 + 369 = 625 = 54

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From the tabulation above

one can deduce that in general

n to the 4th. power is

the summation of triangular numbers

per the formula shown below:

n4 =

n-1

Σ

k=0

(2k + 1)**[**2k(k+1) + 1**]**