n                   a                             b

2 1.61803398874986 1.59349017200339

3 1.32471795724474 1.32018657085151

4 1.22074408460576 1.21918103215648

5 1.16730397826141 1.16658863618974

6 1.13472413840152 1.13433879215268

7 1.1127756842787 1.11254479548739

8 1.09698155779856 1.09683243540866

9 1.08507024549145 1.08496840637825

10 1.07576606608684 1.07569345247076

11 1.06829718892084 1.0682436045058

12 1.06216916786425 1.06212850287508

13 1.05705057522123 1.05701898816211

14 1.05271092014756 1.0526858977005

15 1.04898493475703 1.04896477664951

16 1.04575102415634 1.04573454686422

17 1.04291773230179 1.04290409146387

18 1.04041494778185 1.04040352793187

19 1.03818801943645 1.03817836339662

20 1.03619371713068 1.03618547956491

21 1.03439739613381 1.03439031225564

22 1.03277096644104 1.03276483047836

23 1.03129141247925 1.03128606256801

24 1.02993969667052 1.02993500407428

25 1.02869993552825 1.0286957968237

26 1.02755877239302 1.02755510372435

27 1.02650489414627 1.02650162694533

28 1.02552865476458 1.02552573252459

29 1.02462177913658 1.02461915495304

30 1.02377712786146 1.02377476253873

31 1.02298850886629 1.02298636944579

32 1.02225052531784 1.02224858392397

33 1.02155845192435 1.02155668485204

34 1.02090813363081 1.0209065206186

35 1.02029590211647 1.02029442576322

36 1.01971850654856 1.01971715184225

37 1.01917305583105 1.01917180976573

38 1.0186569701822 1.01865582144466

39 1.01816794032867 1.01816687904086

40 1.01770389295441 1.01770291046034

41 1.01726296131337 1.0172620500021

42 1.01684346012768 1.01684261328548

43 1.01644386405942 1.01644307574581

44 1.01606278917622 1.01606205411948

45 1.0156989769359 1.01569829044476

46 1.0153512802996 1.01535063818817

47 1.01501865165052 1.01501805017435

48 1.01470013225015 1.01469956805152

49 1.01439484300848 1.01439431306945

50 1.01410197638097 1.01410147798294

51 1.01382078923512 1.01382031992389

52 1.0135505965538 1.01355015410929

53 1.01329076586305 1.01329034827305

54 1.01304071228904 1.01304031772622

55 1.01279989416267 1.01279952096461

56 1.01256780910243 1.01256745575407

57 1.01234399051586 1.01234365563408

58 1.01212800446829 1.01212768678838

59 1.01191944687473 1.01191914523842

60 1.01171794097656 1.01171765432146

61 1.01152313507001 1.01152286242022

62 1.01133470045741 1.01133444091523

63 1.01115232959636 1.01115208233487

64 1.01097573442468 1.01097549868116

65 1.01080464484206 1.01080441991209

66 1.0106388073315 1.01063859256371

67 1.01047798370589 1.01047777849722

68 1.0103219499665 1.01032175375796

69 1.01017049526198 1.01017030753486

70 1.01002342093775 1.01002324121019

71 1.00988053966664 1.00988036749049

72 1.00974167465284 1.00974150961093

73 1.00960665890212 1.00960650060566

74 1.00947533455179 1.00947518263815

75 1.00934755225501 1.00934740638559

76 1.00922317061419 1.00922303047255

77 1.00910205565901 1.00910192094915

78 1.00898408036507 1.00898395080988

79 1.00886912420954 1.00886899954933

80 1.00875707276039 1.00875695275164

81 1.00864781729645 1.0086477017106

82 1.00854125445554 1.00854114307788

83 1.0084372859083 1.00843717853701

84 1.00833581805554 1.00833571450076

85 1.00823676174723 1.00823666183019

86 1.00814003202117 1.00813993557346

87 1.00804554786 1.00804545472285

88 1.00795323196485 1.00795314198847

89 1.00786301054444 1.0078629235874

90 1.00777481311832 1.00777472904703

91 1.00768857233328 1.00768849102144

92 1.00760422379178 1.00760414511994

93 1.0075217058916 1.0075216297467

94 1.00744095967578 1.00744088595075

95 1.00736192869214 1.00736185728552

96 1.00728455886168 1.00728448967727

97 1.00720879835513 1.00720873130167

98 1.00713459747721 1.00713453246807

99 1.00706190855788 1.0070618455108

100 1.00699068585024 1.00699062468705

Starting with n=2, where a = phi, the golden ratio, b is somewhat less. As n gets larger, the difference decreases as both seem to approach 1. This seems to be true regardless of the seed value chosen for a in the following program in which the solutions are found iteratively (successively better approximations):

 For n = 2 To 100

   a = 10

   Do

     pa = a

     a = (a + 1) ^ (1 / n)

   Loop Until Abs(pa - a) / a < 0.0000000000001

 

   b = 10

   Do

     pb = b

     b = (b + 3 * a) ^ (1 / (2 * n))

   Loop Until Abs(pb - b) / b < 0.0000000000001

 

   Text1.Text = Text1.Text & n & Str(a) & Str(b) & crlf

 Next n

At around n=24,000, a and b are close enough that they look identical to the precision of the computer (about 15 significant figures).

22500 1.00003080770052 1.00003080770051

22600 1.00003067137815 1.00003067137815

22700 1.00003053625691 1.0000305362569

22800 1.00003040232098 1.00003040232098

22900 1.00003026955485 1.00003026955484

23000 1.00003013794324 1.00003013794324

23100 1.00003000747118 1.00003000747117

23200 1.00002987812391 1.0000298781239

23300 1.00002974988696 1.00002974988695

23400 1.00002962274609 1.00002962274608

23500 1.00002949668731 1.0000294966873

23600 1.00002937169686 1.00002937169686

23700 1.00002924776122 1.00002924776122

23800 1.0000291248671 1.00002912486709

23900 1.00002900300141 1.00002900300141

24000 1.00002888215131 1.00002888215131

24100 1.00002876230415 1.00002876230415

24200 1.0000286434475 1.00002864344749

24300 1.00002852556912 1.00002852556912

24400 1.00002840865699 1.00002840865699

24500 1.00002829269928 1.00002829269928

24600 1.00002817768435 1.00002817768435