ࡱ> Z\Y_tucde7 0bjbjUU R7|7|lTTT$x> > > P $ 4x= (  $ QT X T ^T, *HOx> t,T 0=$},xxGRUPUL ^COLAR DE CHIMIE INDUSTRIAL TG. MURE^ TRIUNGHIUL LUI PASCAL PROF. RUSU-MARIAN CRISTINA TG. MURE^ 2001 Numerele din figura (1) sunt coeficiencii binomiali, iar dispunerea lor sub form de tabel triunghiular se nume_te triunghiul lui Pascal. nsu_i Pascal numea acest triunghi aritmetic. Fig. (1) La triunghiul din figura (1) pot fi adugate noi linii, el poate fi extins orict de mult. Receaua din figura (2) este de fapt, o porciune ptrat  tiat dintr-un triunghi mai mare.  Figura (2) Unii dintre coeficiencii binomiali _i descompunerea lor ntr-un tabel triunghiular apar _i n scrierile altor autori, anterioare lucrrii lui Pascal. Meritele lui Pascal n aceast descoperire sunt suficiente pentru a justifica utilizarea numelui lui. n primul rnd trebuie s introducem o notacie pentru numerele concinute n triunghiul lui Pascal. Pentru noi fiecare numr asociat unui punct din acest triunghi are o semnificacie geometric: el indic numrul de trasee distincte, n zigzag, de lungime minim, de la vrful triunghiului pn la punctul respectiv. Fiecare din aceste trasee trece de-a lungul unui aceluia_i numr de cvartale  s spunem de-a lungul a n cvartale. Mai mult, toate aceste trasee concord ntre ele _i n ceea ce prive_te numrul de cvartale strbtute mergnd spre sud-vest _i numrul de cvartale strbtute mergnd spre sud-est. Fie l _i respectiv r aceste numere (l  nseamn deplasri spre stnga, r  nseamn deplasri spre dreapta, binenceles n fiecare caz direccia general este de sus n jos). Evident: n=l+r. Dac notm dou din cele trei numere n, l _i r, al treilea este complet determinat, _i tot a_a este _i punctul la care ele se refer. Vom nota cu Crn (combinri de n luate cte r) numrul de trasee minime de la vrful triunghiului lui Pascal pn la punctul specificat de numrul n (numrul total de cvartete) _i numrul r (cvartetele strbtute mergnd spre dreapta). De exemplu n figura (3): C38=56; C510=252. Simbolurile pentru numerele din figura (1), au fost grupate n mod corespunztor n figura (3). Simbolurile cu acela_i numr  inferior n se aliniaz pe orizontal n lungul  bazei de ordinul n, este vorba de baza unui triunghi dreptunghic. Simbolurile cu acela_i numr  superior r se aliniaz oblic n lungul  bulevardului cu numrul r.  Figura (3) n al doilea rnd pe lng aspectul geometric, triunghiul lui Pascal prezint _i un aspect legat de proprietci numerice _i de calcul. Toate numerele de-a lungul frontierelor (strada zero, bulevardul zero _i punctul lor comun de plecare) sunt egale cu 1. Prin urmare: C0n=Cnn=1. Aceast relacie se nume_te condicia la limit a triunghiului lui Pascal. Orice numr din interiorul triunghiului lui Pascal este situat pe un anumit rnd orizontal, sau pe o anumit  baz . Un numr oarecare de pe baza (n+1) se calculeaz  mergnd napoi sau  recurgnd la cele dou numere vecine de pe baza n: Crn+1=Crn+Cr-1n. Aceast formul se nume_te formula de recurenc a triunghiului lui Pascal. Din punctul de vedere al proprietcilor de calcul, numerele Crn sunt determinate de formula de recurenc _i de condicia la limit a triunghiului lui Pascal. Cnd calculm un numr din triunghiul lui Pascal folosind formula de recurenc, trebuie s ne bazm pe cunoa_terea prealabil a dou numere de pe baza  precedent . Exist ns o schem de calcul care este independent de cuno_tincele prealabile _i o vom numi formula explicit a coeficiencilor binomiali:  Tratatul lui Pascal concine formula explicit, Pascal nu spune ns cum a descoperit-o dar n schimb d o demonstracie cu totul remarcabil a formulei explicite. n demonstracie Pascal utilizeaz dou leme, n prima lem arat c formula explicit este valabil _i pentru prima linie iar n cea de-a doua lem arat c dac formula este valabil pentru o baz oarecare n, atunci ea este valabil _i pentru baza imediat urmtoare (n+1). Pascal spunea:  Vedem deci c propozicia este, n mod necesar, valabil pentru toate valorile lui n. Cci ea este valabil pentru n=1, n virtutea primei leme, prin urmare ea este valabil _i pentru n=2, n virtutea lemei a doua; prin urmare ea este valabil _i pentru n=3, n virtutea aceleia_i leme _i a_a mai departe, ad infinitum. Cuvintele lui Pascal citate aici au o importanc istoric, fiindc demonstracia dat de el constituie primul exemplu de aplicare a unei scheme fundamentale de racionament, care se nume_te n mod obi_nuit: induccie matematic. Pn acum am dat trei moduri distincte de abordare a numerelor din triunghiul lui Pascal: interpretarea geometric (un coeficient binomial este numrul de drumuri distincte minime, ntre dou noduri ale unei recele de strzi); abordarea formal (adic exclusiv prin calcul, coeficiencii binomiali pot fi definici prin formula lor de recurenc _i prin condicia la limit); formula explicit; Denumirea numerelor ne mai aminte_te o cale: teorema binomului: Pentru orice x (fix sau variabil) _i pentru orice ntreg nenegativ n, are loc egalitatea: Exist _i alte moduri de a aborda numerele din triunghiul lui Pascal, numere ce joac un rol important n foarte multe probleme interesante _i se bucur de foarte multe proprietci interesante.  Acest tabel de numere are proprietci eminente _i admirabile spunea Jaques Bernoulli,  n el st esenca combinatoricii, iar cei familiarizaci cu geometria _tiu c n el sunt ascunse secrete capitale din toat matematica . 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