The tables of partial imaginary tuples listed in Part IB, Part IC,
Part IIIC, Part ID and Part IE
are useful, when squared, as right diagonals for magic square of squares.
I will show that the averages of each of two tuple numbers from the designated tables may be used to generate the next number of averages
in the geometric progressions via successive multiplications with the common ratio R
=
Thus, for example we can generate the geometric progression:
where x_{avg} is a placeholder for a_{avg}, b_{avg}, or c_{avg}. I will show that starting with the initial x_{avg} and gradually replacing the starting x_{avg} with the next real value, there comes a point where the numbers take on the real values. From the tables below, using seven columns and ten rows, the seventh column appears to be that point. Increasing the number of rows, however, produces numbers which are initially far from the real values. To get these numbers more in line with the real values requires us to increase the number of columns as far as is required.
Three tables are generated one for a_{avg} (Table GP_{1}),
one b_{avg} (Table GP_{2}) and one for
c_{avg} (Table GP_{3}).
The sequences for these table were obtained from
where the last underlined number was obtained by multiplying 542868888 by R and not from a table. The numbers are arranged in columns and the initial number averages are multiplied by R over and over until the end of the column. Going across the rows we can see that the numbers approach the real number averages as the numbers get bigger and bigger.
a_{avg}i | 72i | 408i | 2376i | 13848i | 80712i | 470424i | 2741832i |
---|---|---|---|---|---|---|---|
ai_{avg} × R | 419.6i | 2378i | 13848.3i | 80712i | 470424i | 2741832i | 15980568i |
ai_{avg} × R^{2} | 2445.9i | 13860i | 80714i | 470424.4i | 2741832i | 15980568i | 93141576i |
ai_{avg} × R^{3} | 14255.6i | 80782i | 470436i | 2741834i | 15980568.4i | 93141576i | 542868888i |
ai_{avg} × R^{4} | 83088i | 470831.6i | 2741902i | 15980580i | 93141578i | 542868888.3i | 3164071752i |
ai_{avg} × R^{5} | 484272i | 2744208i | 15980975.7i | 93141646i | 542868900i | 3164071754i | |
ai_{avg} × R^{6} | 2822544i | 15994416i | 93143952i | 542869295.6i | 3164071822i | ||
ai_{avg} × R^{7} | 16450992i | 93222288i | 542882736i | 3164074128i | |||
ai_{avg} × R^{8} | 95883408i | 543339312i | 3164152464i | ||||
ai_{avg} × R^{9} | 558849456i | 3166813584i | |||||
ai_{avg} × R^{10} | 3257213328i |
The sequence for b_{avg} of tuple (a_{avg}i,b_{avg}, c_{avg}) is as follows:
where the last underlined number was obtained by multiplying 111960996 by R and not from a table, just as was done above.
b_{avg} | 14 | 84 | 490 | 2856 | 16646 | 97020 | 565474 |
---|---|---|---|---|---|---|---|
b_{avg} × R | 81.6 | 489.6 | 2856 | 16646 | 97020 | 565474 | 3295824 |
b_{avg} × R^{2} | 475.6 | 2853.5 | 16645.6 | 97020 | 565474 | 3295824 | 19209470 |
b_{avg} × R^{3} | 2772 | 16631.6 | 97017.5 | 565473.6 | 3295824 | 19209470 | 111960996 |
b_{avg} × R^{4} | 16156 | 96936 | 565459.5 | 3295821.5 | 19209469.6 | 111960996 | 652556506 |
b_{avg} × R^{5} | 94164 | 564984 | 3295740 | 19209455.6 | 111960993.5 | 652556505.6 | |
b_{avg} × R^{6} | 548828 | 3292968 | 19208980 | 111960912 | 652556491.6 | ||
b_{avg} × R^{7} | 3198804 | 19192824 | 111958140 | 652556016 | |||
b_{avg} × R^{8} | 18643996 | 111863976 | 652539860 | ||||
b_{avg} × R^{9} | 108665172 | 651991032 | |||||
b_{avg} × R^{10} | 633347036 |
The sequence for c_{avg} of tuple (a_{avg}i,b_{avg}, c_{avg}) is as follows:
where the last underlined number was obtained by multiplying 97022475 by R and not from a table, just as was done above.
c_{avg} | 75 | 425 | 2475 | 14425 | 84075 | 490025 | 2856075 |
---|---|---|---|---|---|---|---|
c_{avg} × R | 437.1 | 2477.1 | 14425.4 | 84075.1 | 499025 | 2856075 | 16646425 |
c_{avg} × R^{2} | 2547.8 | 14437.5 | 84077.1 | 490025.4 | 2856075 | 16646425 | 97022475 |
c_{avg} × R^{3} | 14849.6 | 84147.9 | 490037.5 | 2856077.1 | 16646425.4 | 97022475 | 565488425 |
c_{avg} × R^{4} | 86550 | 490449.6 | 2856147.9 | 16646437.5 | 97022477.1 | 565488425.4 | 3295908077 |
c_{avg} × R^{5} | 504450 | 2858550 | 16646849.6 | 97022547.9 | 565488437.5 | 3295908077 | |
c_{avg} × R^{6} | 2940150 | 16660850 | 97024950 | 565488849.6 | 3295908148 | ||
c_{avg} × R^{7} | 17136450 | 97106550 | 565502850 | 3295910550 | |||
c_{avg} × R^{8} | 99878550 | 565978450 | 3295992150 | ||||
c_{avg} × R^{9} | 582134850 | 3298764150 | |||||
c_{avg} × R^{10} | 3392930550 |
The Table GP_{c} show that as we go across columns in diagonal fashion, the numbers approach and eventualy equal the real value. For example, at position
This concludes Part IVF using the eighth rows of two tables for geometric progressions. To see a two novel methods via geometric progression and recursion Part IVF.
Finally to see what started it all go to Part IG which lists the new set of magic square of seven squares.
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Copyright © 2016 by Eddie N Gutierrez. E-Mail: enaguti1949@gmail.com