(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 6
& the resulting product
increased by 1.
6(0) + 1 = 1
6(1) + 1 = 7
6(3) + 1 = 19
6(6) + 1 = 37
6(10) + 1 = 61
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(3)
The tabulation below shows
all the sub-totals of
the numbers per (2),
resulting in all whole numbers
to the 3rd. power.
1 = 1 = 13
1 + 7 = 8 = 23
1 + 7 + 19 = 27 = 33
1 + 7 + 19 + 37 = 64 = 43
1 + 7 + 19 + 37 + 61 = 125 = 53
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From the tabulation above
one can deduce that in general
n to the 3rd. power is
the summation of triangular numbers
per the formula shown below:
n3 =
n-1
Σ
k=0
[ 3k(k = 1) ] + 1