Picture of a square

Intro to Geometric Progressions using Table of Tuples and the Magic Ratio (R)

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 = (1 + √2)2 = 5.8284268.....

Thus, for example we can generate the geometric progression:

xavg × R, xavg × R2, xavg × R3, xavg × R4, xavg × R5, ...

where xavg is a placeholder for aavg, bavg, or cavg. I will show that starting with the initial xavg and gradually replacing the starting xavg 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.

Generation of Tables of Geometric Progressions

Three tables are generated one for aavg (Table GP1), one bavg (Table GP2) and one for cavg (Table GP3). The sequences for these table were obtained from Part IIIF of the previous page which were calculated from the 8th row of tuples for each of Tables VI to XXIV. The sequence for aavg of tuple (aavgi,bavg, cavg) is as follows:

72, 408, 2376, 13848, 80712, 470424, 2741832, 15980568, 93141576, 542868888, 3164071822

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.

Table GPa
aavgi 72i 408i 2376i 13848i 80712i 470424i 2741832i
aiavg × R 419.6i2378i 13848.3i80712i 470424i2741832i 15980568i
aiavg × R2 2445.9i13860i 80714i470424.4i 2741832i15980568i 93141576i
aiavg × R3 14255.6i80782i 470436i2741834i 15980568.4i93141576i 542868888i
aiavg × R4 83088i470831.6i 2741902i15980580i 93141578i542868888.3i 3164071752i
aiavg × R5 484272i2744208i 15980975.7i93141646i 542868900i3164071754i  
aiavg × R6 2822544i15994416i 93143952i542869295.6i 3164071822i  
aiavg × R7 16450992i93222288i 542882736i3164074128i    
aiavg × R8 95883408i543339312i 3164152464i    
aiavg × R9 558849456i3166813584i      
aiavg × R10 3257213328i       

The sequence for bavg of tuple (aavgi,bavg, cavg) is as follows:

14, 84, 490, 2856, 16646, 97020, 565474, 3295824, 19209470, 111960996, 652556506

where the last underlined number was obtained by multiplying 111960996 by R and not from a table, just as was done above.

Table GPb
bavg 1484 4902856 1664697020 565474
bavg × R 81.6489.6 285616646 97020565474 3295824
bavg × R2 475.62853.5 16645.697020 5654743295824 19209470
bavg × R3 277216631.6 97017.5565473.6 329582419209470 111960996
bavg × R4 1615696936 565459.53295821.5 19209469.6111960996 652556506
bavg × R5 94164564984 329574019209455.6 111960993.5652556505.6  
bavg × R6 5488283292968 19208980111960912 652556491.6  
bavg × R7 319880419192824 111958140652556016    
bavg × R8 18643996111863976 652539860    
bavg × R9 108665172651991032      
bavg × R10 633347036       

The sequence for cavg of tuple (aavgi,bavg, cavg) is as follows:

75, 425, 2475, 14425, 84075, 490025, 2856075, 16646425, 97022475, 565488425, 3295908075

where the last underlined number was obtained by multiplying 97022475 by R and not from a table, just as was done above.

Table GPc
cavg 75425 247514425 84075490025 2856075
cavg × R 437.12477.1 14425.484075.1 4990252856075 16646425
cavg × R2 2547.814437.5 84077.1490025.4 285607516646425 97022475
cavg × R3 14849.684147.9 490037.52856077.1 16646425.497022475 565488425
cavg × R4 86550490449.6 2856147.916646437.5 97022477.1565488425.4 3295908077
cavg × R5 5044502858550 16646849.697022547.9 565488437.53295908077  
cavg × R6 294015016660850 97024950565488849.6 3295908148  
cavg × R7 1713645097106550 5655028503295910550    
cavg × R8 99878550565978450 3295992150    
cavg × R9 5821348503298764150      
cavg × R10 3392930550       

The Table GPc show that as we go across columns in diagonal fashion, the numbers approach and eventualy equal the real value. For example, at position (cavg × R6) 2940150 increases to 2856075 at column 5 and remains constant up to the 1strow, 7th column, the real value.

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