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Which is greater? (Posted on 2016-08-28) Difficulty: 3 of 5
Each of A and B is a positive real number and N is an integer with N > 1 satisfying:
AN - A - 1 = 0, and:
B2N - B – 3A = 0

Which of A and B is greater?

No Solution Yet Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

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Solution computer solution (spoiler) | Comment 1 of 3
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


  Posted by Charlie on 2016-08-28 10:29:43
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