# Problem Set 6 Solutions - Problem Set 6 Solutions Due:...

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ProblemSet6SolutionsDue:Monday,March28at9PMProblem1.SammytheSharkisafinancialserviceproviderwhooffersloansonthefol-lowingterms.• Sammyloansaclientmdollarsinthemorning.ThisputstheclientmdollarsindebttoSammy.• Eachevening,Sammyfirstchargesa“servicefee”,whichincreasestheclient’sdebtbyfdollars,andthenSammychargesinterest,whichmultipliesthedebtbyafactorofp.Forexample,ifSammy’sinterestratewereamodest5%perday,thenpwouldbe1.05.(a)Whatistheclient’sdebtattheendofthefirstday?
(b)Whatistheclient’sdebtattheendofthesecondday?
Problem2.Findclosed-formexpressionsequaltothefollowingsums.Showyourwork.
(c)Writeaformulafortheclient’sdebtafterddaysandfindanequivalentclosedform.
2ProblemSet6(a)n9i7i11ii=0Solution.Splittheexpressionintotwogeometricseriesandthenapplytheformulaforthesumofageometricseries.nnin9i7i97i=11i1111i=0i=0i=09n+17n+1111111=9111n+17111n+111911711=++2·114·114(b)n34i+5i=1Solution.Takingthelogarithmreducesthisproducttoaneasysum.n3log3(Qn34i+5)34i+5=i=1i=1Pn= 34i+5i=1= 32n(n+1)+5n(c)n� �1ij5/3·12j1/3j=1i=0Solution.Thisfearsome-lookingsumisapapertiger;wejustapplytheformulaforthesumofageometricseriesfollowedbytheformulaforthesumofanarithmeticseries.n� �1in1j5/3·12j1/3=j5/3·1j=1i=0j=1112j1/3n=2j2j=112n(n+2)(n+ 1)=3
3ProblemSet6

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Tags
Math, Orders of magnitude, Characteristic polynomial, Geometric progression, Recurrence relation, Sammy rst