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Chapter5t
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&ariseoftheirownagproblemsareimportantaeigated.HereIwilldescribethebis，andthealan，FibonadStirliheyeainnaturals.Butwefirstbeginwithsomeveryfualnumbersequences.
Triangularhmetietricprogressions
&heyearwhebis，Iwilltakeamomehetriangularhenthofwhiotedbytn，isdefihefirstnumbers.Itsvalue，intermsofn，befoundbythefollowingtrick.Wewritetmehenagainasthesamesumbutinthereversethetwoversionsofther:
tn=1+2+3+…+（n-2）+（n-1）+n
tn=n+（n-1）+（n-2）+…+3+2+1;
Forexample，bytakinga=1athesumofthefirstnoddnumbersisn+n（n-1）=n+n2-hsquare.
Ifwereplaultipliastheoperation，wemovefromarithmeticseriestogeometricseries.Inaicseries，eachpairofsuccessivetermsisseparatedbyaohenumberbinournotation.Inotherwords，tomovefromohe，wesimplyaddb.Iricseries，weonbeginwithsomearbitraryhefirsttermandmovefromohebymultiplyingbyafixednumber，calledtheonratio，dehesymbolr.Thatistosay，thetypietricserieshastheforma，ar，ar2，…withthenthtermbeingarn-1.Aswitharithmeticseries，thereisaformulaforthesumofthefirstntermsofageometricseries:
Thequickwayofseeingthatthisformulaisrightisto taketheequivalentformthatweobtaiiplybothsidesofthisequationby（r-1）andmultiplyoutthebratheleft-haain:
（ar+ar2+ar3+…+arn）-（a+ar+ar2+…+arn-1）
andthewholeexpressionteleseaningthatermiscelledbyoherbracket:theoionsarearn-a=a（rn-1），showingthatourformulaforthesumiscorreple，puttinga=1andr=2givesusthesumofpowersof2:
1+2+4+…+2n-1=2n-1.
ThisformulaisjustwhatyouneedioverifyEuclid’sresultfromChapter3oeevenumbersfromMersenneprimes.
Factorials，permutations，andbis
Aswehaveseeriangularnumberarisesfromsummingallthenumbersfrom1uptoher.Ifwereplaultiplithisidea，wegetwhatareknoworialnumbers，whichmadetheirfirstappearaer2.
Faeuptlyingandprobabilityproblemssuchasthecesofbeiatypeofhandinacardgamelikepoker.Forthatreasoheirownnotatioorialisdenotedbyn!=n×（n-1）×…×2×1.Thetriangularnumbersgrowreasonablyquickly，atabouthalftherateofthesquares，butthefactrowmuchfasterandsoonpassintothemillionsandmillions:forexample10!=3，628，800.Theexarkalertsustothisratheralarmih.
&specialclassthatemergesingproblems，oreionsastheyarecalled，isthatofthebis，soheyariseasthemultipliersofpowersofxwhenthebinomialexpression（1+x）nisexpahebi，r）isthenumberofdifferentwayswemaystructasetofsizerfromoneofsizen.Forexample，C（4，2）=6，astherearesixpairs（takenwithardtoorderair）thatagroupoffour:forexample，ifwehavefour，Alex，Barbara，e，andDavid，therearesixwaysthatweselectapairfromthisgroup:AB，AC，AD，BdCD.
Thebisbecaltwodistinctways.First，wedtheargumeweusedtocalculatege，r）r!，whisgivesustheusefulexpression:
Anotablespecialcaseiswher=2，whichds
tothenumberofpairsthatbeseleasetofs.
Thisfactorial-basedformulaforgbisdoesgiveanicealgebraithebisthatallowsustodemoheirmanyspecialproperties.However，theevolutioiesisofteraifwefoasedwaytogeegers，whichisbymeaigle（seeFigure2），alsoknoascal&#le，inhonourofthe17th-turyFreidphilosopherBlaisePascal（1623–62）.（TheArithmetiglehasbeendisdre-dischoutPersia，India，andaoverthelast1，000years:forexample，itfeaturedasthefrohePreirrorbyChuShih-1303.）
2.TheArithmetigle
Eaumberinthebodyleisthesumofthetwoaboveit.Thetriangle，whibeuediely，givesthefulllistofbis.Forexample，tofindthenumberofwaysofselegfiveperoupofseven，proceedasfollows.heliriangle，beginning.Similarlyionswithinealefttainstartingwith0.Godowntothelinenumbered7，ahelinenumbered5（rememberingtostartyour0）:weseetheansweris21.Youwillryofeaple，21isalsothenumberofwaysofgtwoperoupofseven.Thisisexplainedbythatwhehefivefromseveaneouslygtwofromsevewobeingthepairleftbehind.Thissymmetryargumentofcourseappliestoeveryrow.Thisisalsomaheformulaonpage55，foritreturnsthesameexpressionifwerepla-r，asthetermsrandn-rthatweseeiorsimplysositions.
&hatthepatterherightahardtosee.Eachrowbuildsfromtheo.Weseeeasilythatthefirstthreerowsareple，the2ihethirdrowtellsusthattherearetwowaysofgasinglepersonfromapair.The1thatsitsontopissayingthatthereisoochooseasetofsizezerofromtheemptyset.Infact，thereisonewayofgasetofsizezerofroma，whichiswhyeveryrowbeginswith1.Letusfotheexamplejustgiven–thereare21=15+6waysofselegfivefromagroupofsevehe21quiurallysplitintotwotypes.First，thereare15waystroupoffourfromthefirstsixpeople，towhichwemayaddtheseveoformourfivesome.Ifwedon’tihpersoheobuildasetoffivefromthefirstsix，andtherearesixwaysofdoingthis.Thisillustrateshowoheryisthesumofthetwoaboveit，andthispatternpropagatesthroughle.Insymbolsthisruletakestheform:
，r）=-1，r）+-1，r-1）.
&riangleisripatterns.Forexample，summingallthenumbersineachsuccessiverowgivesthedoublingsequence1，2，4，8，16，32，…:thesequenceofpowersof2.Insummibegins1，8，28，56，…forinstance，wearesummingthenumberofwaysofgasetofsize0，1，2，3etasetof8.Intotal，thisgivesusthenumberofwaysofselegasetofanysizefromagroupof8，whichisequalto28as，ingeneral，asetofsizens2hinit.
Thislastfabeseely，forasubsetofasetofsizeifiedbyabinarystrihninthefolloesiderthesetiioninaspecificorder{a1，a2，…，an}say，andthenabinarystrihnspecifiesasubsetbysayianthestringihepresehedingaiiiion.Forexample，ifrings0111and0000staivelyfor{a2，a3，a4}，
ayset.Siwochoicesforeatryinthebinarystring，thereare2nsugsinallandtherefore2hiofsizen.
umbers
&hesimplestvisualrepresentationsthatgivesrisetothisypeisasthenumberofwayswedraw‘mountains’usingnupstrokesandndownstrokes（seeFigure3）
3.Withthreeupanddowherearefivemountainpatterns
&ainpatternhasaion，however，asameaningfulbradsothenumberofmeaningfulwaysaofnpairsofparehenthumber.Forexample，（（））（）and（（（）））aremeaningfulbragsbut（））（（）isnot:tobemeaningful，thebracketsmustneverfallbehindthenumberhtbracketsaswelefttht.Thisdstothenaturalthatourmountainsmustneverdiveunderground.ForihefirstandlastmountainpatternsinFigure3dt（）（（））aively.
&alannumberalsotsthenumberofwaysthatwebreakuparegularpolygonwithrianglesbymeansofdiagonalsthatdonoteahereareotheriionsalongtheselihbis，therearefumberstosmallerumbers，whichmakesthemameomanipulation.
Fibonaumbers
TheFibonaeseriesofengeionamongthegeneralpublic.Thesequensasfollows
1，1，2，3，5，8，13，21，34，55，89，144，233，377，610，…
whereeaumberafterthepairofinitial1sisthesumofthetwothatebefore.Inthis，thereisasimilaritywiththebisiermisthesumoftwopreviousohesequehemethodofformationoftheFibonaumbersissimpler:
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