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 Powering Up the Root (Posted on 2011-09-24)
A sequence {A(n)} satisfies:

A(1) = √2, and:

A(n)= (√2)A(n-1) for n =2, 3,……

Determine this limit:

Limit A(n)
n → ∞

 No Solution Yet Submitted by K Sengupta Rating: 2.0000 (1 votes)

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 computer solution Comment 3 of 3 |

Iterating we see it approaching 2. Also, as an iterative way to solve a=(sqrt(2))^a, it makes sense that it does approach 2, as that is the solution to this equation.

`2       1.632526919438152844773495381024719602079108857033       1.760839555880028090756649895638372748079804094284       1.8409108692910102984148345409764592614995045817035       1.8927126968285104808234546164831610418439565001516       1.9269997018471004310156383175181077007005453516887       1.9500347738058175818287432696203262685259166881188       1.9656648865173187138901476127422961715664757695599       1.9763417544097023776963406248777850817858304678810      1.98366839930382114719375174612029457251318871737611      1.98871177341395331832810079339031012901511588757612      1.99219088294705709975430514843794217813745208084713      1.99459445071210117762895413462800996669843851878714      1.99625666626585838379696172198427049924965564973515      1.99740700114133583223726944496367774743009178928316      1.99820347750870162388988290083805268958814211840917      1.9987551330845918851370073345028940880734522969518      1.99913731011939093325132521985847820930944293707719      1.99940211832499657289040766090102301092844069822720      1.99958562293568103044002687648814544303096951503221      1.999712796329640067761597962489529531981825778122      1.99980093549297043926745415842790836436354135308423      1.99986202375778266290398486557359003751489601123924      1.9999043644433356154422325966293903271543717144625      1.99993371158209966195349755743359004724351611105426      1.99995405289782073446314855583339607332000339620827      1.99996815214924363375776965666109836359473320157128      1.99997792487387002569316650818982160131406644007929      1.99998469874709488678876566615875684963529125137630      1.99998939400781165310022310949966158140113730283531      1.99999264849992877797541074336569132983945782306332      1.99999490433494420783283239989693581768381205621533      1.99999646795725233498770217175818903945974552748834      1.99999755177602629017358208348258647756079390673635      1.99999830302117517845836082879907086917486607889636      1.99999882374425799954734382389003313234383324108937      1.99999918468181500092065960698535059166297606222638      1.99999943486457865316241544476003578691925328278639      1.99999960827801442041630360049871654267839934788440      1.99999972847902856307915748443190443123845195340441      1.99999981179601304078660812562795475121686819520242      1.99999986954694132358083720511995487061975976570443      1.99999990957683222710632563794764781791204568378644      1.99999993732343718300774267508472506115437509520745      1.99999995655591766806059264270156056196676462272146      1.999999969886857046302412648721800431606909361955`
` 5   point 1010   A=sqrt(2)20   for Iter=2 to 4630    A=sqrt(2)^A:print Iter,A40   next`

 Posted by Charlie on 2011-09-24 14:57:05

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