...a pentru orice kn rezulta pn1 adevarata, pentru orice nk.2. PERMUTARIFie EI1, 2, ,nS o multime finita cu n elemente. Se numeste permutare a multimii E orice functie bijectiva f E E.Notam permutarea in felul urmatorNotam numarul de permutari Pn Pn n!1.2.3nconditie de existenta nNconventie 0!1 1!1Pnnn-1!nn-1n-2!3. ARANJAMENTENotam cu AnkSistemele ordonate cu k elemente, care se pot forma cu elementele unei multimi cu n elemente nk, se numesc aranjamente de n elemente luate cate k. Ankn!n-k!nn-1n-2n-k1n-k1Ank-1c.e. nkconventie nk AnnPn4. COMBINARI Cnkconventie Cn0Cnn1 c.e. nkFormule pentru combinari complementare CnkCnn-k CnkCn-1kCn-1k-15. BINOMUL LUI NETONDaca a, bR, nN, atunciabnCn0anCn1an-1bCn2an-2b2Cnkan-kbkCnn-1abn-1
Cnnbnsau Tk1termen generalkse numeste rangul termenului al dezvoltarii a-bn Cn0an-Cn1an-1bCn2an-2b2--1n-kCnkan-kbk-1n-1Cnn-1ab
n-1-1nCnnbn sauObs 1 in dezvoltarea abn, dupa formula lui Neton, sunt n1 termeni.2 Cn0, Cn1, Cn2,,Cnn se numesc coeficienti binomiali3 Sa se faca distinctie intre coeficientul unui termen al dezvoltarii si coeficientul binomial al aceluiasi termen.4 Pentru a determina rangul celui mai mare termen folosim relatia5 In dezvoltarea abn si a-bn, daca ab atunci Cn0Cn1Cn2Cnn2n Cn0Cn2Cn4Cn1Cn3Cn52n-16 Identitatile utileCnkCn-1k-1Cn-2k-1Ck-1k-1CnkkCn0CmkCn1Cmk-1Cnk
Cm07 Suma puterilor asemenea ale primelor n numere naturaleFie k1 un numar natural si Sk1k2k3knk Folosim dezvoltarea a12a22a1 pentru demonstratie unde a1,2,n.Folosim dezvoltarea a13a33a23a1, pentru demonstratie, unde a1,2,n.Folosim dezvoltarea a14a44a36a24a1, pentru demonstratie, unde a1,2,n Caz particular 6. PROGRESII ARITMETICE SI GEOMETRICE Teorema Fie numerele an-1, an, an1 in progresie aritmetica. Atunci 2anan-1an1Def Fie numerele a1, a2, a3,,an in progresie aritmetica, daca ana1n-1r sau anan-11, unde an ultimul termen a1primul termen an-1penultimul termen nnumarul de termeni rratia progresiei aritmeticeObs Pentru verificare ra2-a1a3-a2a4-a3an-an-1Teorema Fie numerele bn-1, bn, bn1 in progresie geometrica. Atunci bn2bn-1.bn1Def Fie numerele b1, b2,bn in progresie geometrica, daca bnb1.qn sau bnbn-1.q unde bnultimul termenb1primul termenbn-1penultimul termen nnumarul de termeni qratia progresiei geometriceObs pentru verificare qa2a1a3a2anan-1PAGE PAGE 4 EMBED PBrush EMBED PBrush EMBED PBrush EMBED PBrush EMBED PBrush EMBED PBrush EMBED PBrush EMBED PBrush EMBED PBrush EMBED PBrush EMBED PBrush EMBED PBrush EMBED PBrush EMBED PBrush Xxyzyzaaaaaaaaaaajabaaahu,Q5CJmHsHajhu,QUjhu,QCJmH
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