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The Irrational Units Digit (Posted on 2007-05-21) Difficulty: 4 of 5
Let n be a positive integer. Find a formula for the units digit of (11+sqrt(111))^n.

See The Solution Submitted by Brian Smith    
Rating: 3.0000 (1 votes)

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Some Thoughts well, it looks like | Comment 2 of 4 |

The integer parts for n=1 to 44 are:

1       21
2       463
3       9987
4       215095
5       4632231
6       99758143
7       2148356847
8       46266269215
9       996374354271
10      21457573101823
11      462102864697407
12      9951687292324735
13      214316091784170111
14      4615437146328495103
15      99396456301385191167
16      2140567667167189254655
17      46098524114664311690751
18      992761853850942964649983
19      21379775543574102105392127
20      460427443420120816672126975
21      9915605999806916945732872191
22      213539057561550964639401918463
23      4598703206356052052609513484287
24      99036079964217635511015277469695
25      2132806727149227460716240969490431
26      45931387197640827780647148554092543
27      989162451076605936567074858495131647
28      21302260051708922326669175401351970815
29      458758096626830231821051110244792041471
30      9879655525273175876796432671371905204223
31      212764840589741566971311007667733994078207
32      4582029937721582714600877841976428817678335
33      98677010223977404051506202446804094048141311
34      2125073925550287061987127675409925780882325503
35      45764856259866541323201746834550326238929747967
36      985576098461561038490567153606007919447631200255
37      21225025603555677433560459910986670965458588925951
38      457094802293609293153424446505646682045612644368383
39      9843835394423847675039733224014360295348892286844927
40      211993430654388555919339886463259459677219503866904575
41      4565417120452309753475080169951564509945340162203451391
42      98319242343406929017258364874301824622025288529806884863
43      2117369160350429340844933225535124496585102946033716953087
44      45598929104275376208415947313029720678652011927443704119295

and the last digits follow the sequence of 1, 3, 7, 5, 1, 3, 7, 5, ....

The units digits continue the pattern beyond 44:

 1  3  7  5  1  3  7  5  1  3  7  5  1  3  7  5  1  3  7  5  1  3  7  5  1  3  7
  5  1  3  7  5  1  3  7  5  1  3  7  5  1  3  7  5  1  3  7  5  1  3  7  5  1
3  7  5  1  3  7  5  1  3  7  5  1  3  7  5  1  3  7  5  1  3  7  5  1  3  7  5
 1  3  7  5  1  3  7  5  1  3  7  5  1  3  7  5  1  3  7  5  1  3  7  5  1  3  7
  5  1  3  7  5  1  3  7  5  1  3  7  5  1  3  7  5  1  3  7  5  1  3  7  5  1
3  7  5  1  3  7  5  1  3  7  5  1  3  7  5  1  3  7

 10   point 244
 20   A=11+sqrt(111)
 30   T=A
 40   for I=1 to 44
 50     print I,int(T)
 55     T=T*A
 60   next
 70   for I=45 to 300
 80       print int(T)@10;
 90       T=T*A
100   next

which stopped with and overflow at i = 195.

From the fractional parts it looks as though they are asymptotically approaching the next integer in each case, but as it's asymptotic, they would never get there.

1       0.535653752852738848401404661899667476747700659724348330440129127826
2       0.784382562760254664830902561792684488449414513935663269682840812173
3       0.899878852198214142265809740442383978410112709341108628621206589540
4       0.953509120758164481538788671805602640528334466147757132838136848158
5       0.978412134697477171195253375299418307522231161839570636226944764064
6       0.989975755762852950907687538531176360205740898992982668611416327843
7       0.995345279807993208016592094691696849303988159449912347181711571895
8       0.997838598147321067288150697905567082630330517968244951883491303253
9       0.998996361161131400173394407005507324827389800802265469619692952627
10      0.999533964071680130933169975065490319899270437967390812798331925253
11      0.999783597965648878795795381385713789510051627259943185366372829308
12      0.999899514527474024175798639830800170228431420044841950076882992249
13      0.999953339947939743909616262420465849924974968387091048027697536401
14      0.999978333579934124253571374942246996065135104067583555840515878333
15      0.999989939279153294482407624524775414183222605615927748214373959313
16      0.999995328342031236077253990122589151379546282874574902311068321553
17      0.999997830733154248875511537449207188517792167081370368699763481028
18      0.999998992709081114488713922656666633595964847044399088284113367085
19      0.999999532268242029996590923954594053933304964163076255252859265591
20      0.999999782810513515037861100434402850573060741143686732721770172157
21      0.999999899148877030867034970010922173274286663530345567350351131538
22      0.999999953170159528696158335896259306303699186230735154490023172265
23      0.999999978254739322645133689608483005938515461772717725276998474451
24      0.999999989902669811231357812424033067610348296692438411193734715278
25      0.999999995311342620638534977243897428042507909506467793492178991594
26      0.999999997822839541734191375125412740831691042217907344890590662291
27      0.999999998989043711766860480320106017872123833729283652671204654457
28      0.999999999530566241529016815788204984869813919865166909860595775147
29      0.999999999782020195969765144139449488414667899740835490221060508661
30      0.999999999898781896044665013185838896424554595646711686257373439065
31      0.999999999952999753284978848693960837193522106819302195451610572824
32      0.999999999978175611822884539408749454011940393557531437361698211479
33      0.999999999989865927253671380052879616327467590072669667441254924296
34      0.999999999995294281351924967075857019084883046023418310090626219714
35      0.999999999997814917205635475140058256592751111788506147581227590760
36      0.999999999998985365004730782322711454191693999112952145880744799586
37      0.999999999999528858047722459699069426289756862599885733564109683282
38      0.999999999999781227002586290152412836457710986067964679602965036338
39      0.999999999999898413579673786362388139172073067496365615624133966629
40      0.999999999999952828726960398448410697208497624240396747701296902458
41      0.999999999999978096196390902241153946866217058325072293187192187792
42      0.999999999999989829050995864821279858971799040747622973105259106849
43      0.999999999999995277158000003656617428717408313196982476443778472764
44      0.999999999999997806966041432232784842064992482857384750710535332310

So it looks like:

d = (2^n - 1) mod 10


  Posted by Charlie on 2007-05-21 14:56:05
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