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Math Help - Is it true that...

  1. #1
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    Is it true that...



    How would you prove it?
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  2. #2
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    Well, you have 499 terms there, each of them smaller than 1, so LHS < RHS and the proposition is false.
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  3. #3
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    Quote Originally Posted by wonderboy1953 View Post


    How would you prove it?
    You're joking, right?
    That equals ~4.334170287753...
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  4. #4
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    Perhaps you meant:
    SQRT[(1-2)^2] + SQRT[(2-3)^2] ..................... + SQRT[(499-500)^2]

    But if you did mean that, hmmm.....
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  5. #5
    Super Member Bacterius's Avatar
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    Actually it is approximately 4.334170287753113234595879500453267755310454988694 20258271329810004575616000048214823421426367474711 74007491857921898726035780247908644706302248478356 34246364812763384539250878658978385768756347895153 17967788338143571810755723574255971134756890611440 98609730168136987947106502322967494834252498219572 01228578960590617891626405362367025575890551281492 10112514699717127287215834956874432975870074911849 66129936487963753240815996596464134132024452943532 64171525976925204133841620788614763081661139962992 85961196814236742989241961087040027933747518355646 47444390142065474902729908196866777976993828584274 51809742781811073558302786904366466738411510701970 31685725149847273111520810146494899163687973485742 84887208810330002604044691616596633479480512444607 19939508631740012361482872772633822285496051504802 54023185922520200352402006423273342356999690867388 44944863311975256312327570490077905939989130655397 76368477781970241960192610393225330609368750007062 24045686982076661740013924109458440135279484440146 36328111154243374731267924474836151468398392164533 31135316422711682106452788848190911594807001357378 12297626652353903161499725885518253210180645817691 21492371759277788239183154023302643312695964281386 06421881990705857187278751190055819246962031205679 93811560928519525921225396053290801674060680606843 32724058707732783342331904964100393421689020858200 58763616828857889757284807277436758503586940683837 91074470975090364617497369481599314150367329624206 85512973297636642485806224036884736463911105638751 17888179956641243856331038313944019407392915086698 43504984276108110157092744426629544890416797464315 94078739015972746025033722606810577141597592779373 74993668321416150753518702349657247531147838470937 03328244578295147087225493490307299558653750066378 62430836136257756925142423500027602760297228911448 49329541381360003909297047476669830606551482037756 27300787585362212486949702782114119500163516941832 37127252143889688616469826029586414649562454161869 21427184863014624854240725522155666750452602264123 60306425446289540467598307313737518330651113458566 51557340810631343471694138118387779101148547359938 71051647858511685834432801900944682940225108147645 65395989976195292861660849594631853509754899208570 31576313880593099958240976740541112591893228916836 98867729406676446263794704623016743718150118495022 75979233020611618236257597272333541966320935291232 70984396647522747678400134965265037680895877502064 10530499890387744385143631077106538661593634861776 76095002316136452566412346290303867113558236310483 07817921234909095010589255005220807204223712040926 37686442900332366370323943808244045116419739637652 17453163688400010251802105922543207614616233299626 49875379374160354564304159986126890723365879074207 20504395222357238251583742969849308605266744808702 73623647737603182248685133171431566775766762834195 82837772740125887062795923337504804866410494988857 33233425049389625671480601622859677934993206059468 19409082679439342284663687538077505819361220202094 65087301583289030811248220186437772815695322509392 61726393747886068942850512940964522836838504801324 50875676024482933769955528689676775284962687429915 36110440978518987912709727234132210956811568134956 54987556151332227812022969075239314208024634763639 4

    But I doubt this is relevant

    Didn't you mean \sqrt{1 + 2 + 3 + \cdots + 500} = 499 ?


    Hey I can see my house in these numbers !
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  6. #6
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by Bacterius View Post
    Actually it is approximately 4.334170287753113234595879500453267755310454988694 20258271329810004575616000048214823421426367474711 74007491857921898726035780247908644706302248478356 34246364812763384539250878658978385768756347895153 17967788338143571810755723574255971134756890611440 98609730168136987947106502322967494834252498219572 01228578960590617891626405362367025575890551281492 10112514699717127287215834956874432975870074911849 66129936487963753240815996596464134132024452943532 64171525976925204133841620788614763081661139962992 85961196814236742989241961087040027933747518355646 47444390142065474902729908196866777976993828584274 51809742781811073558302786904366466738411510701970 31685725149847273111520810146494899163687973485742 84887208810330002604044691616596633479480512444607 19939508631740012361482872772633822285496051504802 54023185922520200352402006423273342356999690867388 44944863311975256312327570490077905939989130655397 76368477781970241960192610393225330609368750007062 24045686982076661740013924109458440135279484440146 36328111154243374731267924474836151468398392164533 31135316422711682106452788848190911594807001357378 12297626652353903161499725885518253210180645817691 21492371759277788239183154023302643312695964281386 06421881990705857187278751190055819246962031205679 93811560928519525921225396053290801674060680606843 32724058707732783342331904964100393421689020858200 58763616828857889757284807277436758503586940683837 91074470975090364617497369481599314150367329624206 85512973297636642485806224036884736463911105638751 17888179956641243856331038313944019407392915086698 43504984276108110157092744426629544890416797464315 94078739015972746025033722606810577141597592779373 74993668321416150753518702349657247531147838470937 03328244578295147087225493490307299558653750066378 62430836136257756925142423500027602760297228911448 49329541381360003909297047476669830606551482037756 27300787585362212486949702782114119500163516941832 37127252143889688616469826029586414649562454161869 21427184863014624854240725522155666750452602264123 60306425446289540467598307313737518330651113458566 51557340810631343471694138118387779101148547359938 71051647858511685834432801900944682940225108147645 65395989976195292861660849594631853509754899208570 31576313880593099958240976740541112591893228916836 98867729406676446263794704623016743718150118495022 75979233020611618236257597272333541966320935291232 70984396647522747678400134965265037680895877502064 10530499890387744385143631077106538661593634861776 76095002316136452566412346290303867113558236310483 07817921234909095010589255005220807204223712040926 37686442900332366370323943808244045116419739637652 17453163688400010251802105922543207614616233299626 49875379374160354564304159986126890723365879074207 20504395222357238251583742969849308605266744808702 73623647737603182248685133171431566775766762834195 82837772740125887062795923337504804866410494988857 33233425049389625671480601622859677934993206059468 19409082679439342284663687538077505819361220202094 65087301583289030811248220186437772815695322509392 61726393747886068942850512940964522836838504801324 50875676024482933769955528689676775284962687429915 36110440978518987912709727234132210956811568134956 54987556151332227812022969075239314208024634763639 4

    But I doubt this is relevant

    Didn't you mean \sqrt{1 + 2 + 3 + \cdots + 500} = 499 ?


    Hey I can see my house in these numbers !
    How did you do that?!
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  7. #7
    Super Member Bacterius's Avatar
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    Mentally, what were you thinking ...
    Joking, I typed in the sum in Wolfram Alpha and clicked the "More digits" option repeatedly.
    Kinda lame somehow
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  8. #8
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    Sorry I made the problem too hard:

    1/(\sqrt{1} + \sqrt{2}) + 1/(\sqrt{2} + \sqrt{3}) + 1/(\sqrt{3} + \sqrt{4})+...+1/(\sqrt{498} + \sqrt{499}) + 1/(\sqrt{499} + \sqrt{500}) = 499?

    It's a great problem, algebra wise (that's a hint) and how would you go about proving that the left side of the equation equals the right side?
    Last edited by wonderboy1953; June 28th 2010 at 07:59 AM. Reason: further correction
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  9. #9
    Super Member Bacterius's Avatar
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    Uhm ... alright ...

    Well \displaystyle\sum_{k = 1}^{499} \frac{1}{\sqrt{k^2} + \sqrt{\left (k + 1 \right )^2}} = \displaystyle\sum_{k = 1}^{499} \frac{1}{2k + 1}}

    Now there are 499 terms, and each of them is smaller than 1 since 2k + 1 > 1 for any integer k > 0. And the sum of 499 terms all less than 1 cannot possibly be equal to 499, therefore the proposition is false ...

    Can you send a link to this great problem, as the one you are desperately trying to show us is trivial to actually disprove ...
    Last edited by Bacterius; June 28th 2010 at 08:09 AM.
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  10. #10
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    Quote Originally Posted by wonderboy1953 View Post
    Sorry I made the problem too hard:

    1/(\sqrt{1^2} + \sqrt{2^2}) + 1/(\sqrt{2^2} + \sqrt{3^2}) + 1/(\sqrt{3^2} + \sqrt{4^2})+...+1/(\sqrt{498^2} + \sqrt{499^2}) + 1/(\sqrt{499^2} + \sqrt{500^2}) = 499?

    It's a great problem, algebra wise (that's a hint) and how would you go about proving that the left side of the equation equals the right side?
    Doesn't the LHS just boil down to \displaystyle\sum_{i=1}^{499}\frac{1}{2i+1}\approx 3.08905914555508261116187...?

    Edit: too slow...
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  11. #11
    Super Member Bacterius's Avatar
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    Do you want me to get a couple thousand digits ?
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  12. #12
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    Quote Originally Posted by Bacterius View Post
    Do you want me to get a couple thousand digits ?
    Haha, only if you do it mentally.
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  13. #13
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    I'm now more confused than I was!!
    Left side equals 3.0890591....

    The series is simply the reciprocals of odd numbers from 3 to 999.

    Are you feeling ok?!
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  14. #14
    Super Member Bacterius's Avatar
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    Quote Originally Posted by wonderboy1953 View Post
    Sorry I made the problem too hard:

    1/(\sqrt{1} + \sqrt{2}) + 1/(\sqrt{2} + \sqrt{3}) + 1/(\sqrt{3} + \sqrt{4})+...+1/(\sqrt{498} + \sqrt{499}) + 1/(\sqrt{499} + \sqrt{500}) = 499?

    It's a great problem, algebra wise (that's a hint) and how would you go about proving that the left side of the equation equals the right side?
    Same problem ! Each term is smaller than 1 therefore the proposition is false. I join Wilmer on this point : are you feeling all right ? Or does the 1 / a notation mean something else than inverse to you ?
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  15. #15
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    Quote Originally Posted by Wilmer View Post
    I'm now more confused than I was!!
    Left side equals 3.0890591....

    The series is simply the reciprocals of odd numbers from 3 to 999.

    Are you feeling ok?!
    Are you sure about this, Wilmer and Bacterius? Please recheck
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