1

1

1

1

1

1

1

1

1

1

2

3

4

5

6

7

8

9

1

3

6

10

15

21

28

36

45

1

4

10

20

35

56

84

120

165

1

5

15

35

70

126

210

330

495

1

6

21

56

126

252

462

792

1287

1

7

28

84

210

462

924

1716

3003

1

8

36

120

330

792

1716

3432

6435

1

9

45

165

495

1287

3003

6435

12870

Now we can always derive the unique number sequences for (x
– 1)^{n + 1 }from the corresponding sequences for (x –1)^{n} through
application of the fact that the kth term in the former = the sum of the first
k terms in the latter sequence.
Therefore as we can see in the unique number sequence for (x
– 1)^{3 }i.e. 1, 3, 6, 10, …, the first term 1, represents the sum of
the first single term in the corresponding sequence for sequence for (x – 1)^{2
}i.e. 1, 2, 3, 4, …; the second term, 3 then represents the sum of the
first 2 terms in the corresponding sequence i.e. 1 + 2; the 3^{rd}
term, 6 represents the sum of the first 3 terms in the previous sequence i.e. 1
+ 2 + 3; the fourth term, 10 then represents the sum of the first 4 terms in
the previous sequence i.e. 1 + 2 + 3 + 4 and so on indefinitely.
However the limitation of this procedure is that we must
already know the unique number sequence corresponding to (x – 1)^{n }to
be able to calculate the corresponding sequence for (x – 1)^{n + 1}.^{
} ^{ }
However there is a simple way to calculate independently the
unique number sequence corresponding to (x – 1)^{ n} for any given n.
The bais for this calculation is that in general terms the
ratio of the (k + 1)^{th} to the k^{th} term
i.e. (k + 1)^{th}/ k^{th} = (k + n – 1)/k
So for example in the sequences above, when n = 4, the
unique digit sequence for (x – 1)^{4}
is given by 1, 4, 10, 20, 35, 56, 84, 120, 165,…
So if for example we take k = 6 then the ratio of the 7^{th} to the 6^{th}
term i.e. 7^{th}/6^{th} = (6 + 4 – 1)/6 = 9/6 (i.e. 3/2).
And as we can see, this is indeed true for the 7^{th}
term = 84 and the 6^{th} term = 56 and 84/56 = 3/2.
So aided with this simple general fact, regarding the ratio
of successive terms, to illustrate, I will now calculate the unique number
sequence corresponding to (x – 1)^{12}.
Now the first term is  as always  1.
Therefore the 2^{nd}
term =1 * (1 + 12 – 1)/ 1 = 12.
The 3^{rd} term then = 12 * (2 + 12 – 1)/2 = 12 *
13/2 = 78.
The 4^{th} term = 78 * (3 + 12 – 1)/3 = 78 * 14/3 =
364.
The 5^{th} term = 364 * (4 + 12 – 1)/4 = 364 * 15/4 = 1365.
The 6^{th} term = 1365 * (5 + 12 – 1)/5 = 1365 *
16/5 = 4368.
The 7^{th} term = 4368 * (6 + 12 – 1)/6 = 4368 *
17/6 = 12376.
So the unique digit sequence corresponding to (x – 1)^{12}
is
1, 12, 78, 364, 1365, 4368, 12376, …
Now we already know that the 12 roots of this equation are
1.
However if we attempt to approximate these roots through the
ratio of (k + 1)^{th}/k^{th} terms, we must include a great
number of terms so as to get a valid approximation.
So ultimately (k + 1)^{th}/k^{th } term ~
1 (when k is sufficiently large).
I then looked at the sum of reciprocals for these unique
number sequences to find that an interesting general pattern was at work.
Clearly from a conventional perspective, the sums of
reciprocals of the numbers associated with the first two sequences for (x – 1)^{1
}and (x – 1)^{2 }respectively, diverge.^{ }
However the sum of reciprocals of the sequence,
corresponding to (x – 1)^{3}
i.e. 1 + 1/3 + 1/6 + 1/10 + 1/15 + … converges to 2.
In fact a general result can be given for all such
convergent sequences with respect to the sums of reciprocals of all the unique
number sequences associated with (x – 1)^{3} where n ≥ 3.
This result in fact depends solely on the value of n (as the
dimensional power or index) and is given simply as (n – 1)/(n – 2).
So for example, the sum of reciprocals of the next number
sequence, corresponding to
(x – 1)^{4}, i.e.
1 + 1/4 + 1/10 + 1/20 + 1/35 + …, converges to (4 – 1)/(4 –
2) = 3/2.
And the sum of reciprocals corresponding to the number
sequence uniquely associated with
(x – 1)^{12}, that we earlier calculated i.e.
(x – 1)^{12}, that we earlier calculated i.e.
1 + 1/12 + 1/78 + 1/364 + 1/1365 + 1/4368 + 1/12376 + …
thereby converges to (12 – 1)/(12 – 2) =
11/10 = 1.1.
In fact the sum of the first 7 terms above = 1.09994…, which
is already very close to the postulated answer.
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