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Exercise:
Dimostra per induzione completa su k e N che _n^Nnn+n+n+s n+kfrack+NN+N+N+s N+k+

Solution:
footnotehttps://en.wikipedia.org/wiki/Hockey-stick_identityAbbiamo che _nk^Nnchoose kN+ choose k+. dunque riscrivo il primo termine tramite i coefficienti binomiali: *_n^Nnn+n+n+s n+k_n^Nfracn+k!n-! k+! _n^Nfracn+k!n-!k+! k+! _n^Nn+k choose k+ * e via un index shift otteniamo * _n^Nnn+n+n+s n+kk+! _n^Nn+k choose k+ k+!N+k+ choose k+ frack+ N N+s N+k+ * textbfIn alternativa: con induzione su N e poi su k: * _n^N+&nn+n+n+s n+k _n^Nnn+n+n+s n+k+N+N+N+s N+k+ &oversetPIfrack+NN+N+N+s N+k++N+N+N+s N+k+ leftfracNk++rightN+N+N+s N+k+ leftfracN+k+k+rightN+N+N+s N+k+ frack+ N+N+N+s N+k+ N+k+. * Mentre per induzione completa su k * _n^N+&nn+n+n+s n+k n+k+ _n^N+&nn+n+n+s n+k n+k+ *
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Exercise:
Dimostra per induzione completa su k e N che _n^Nnn+n+n+s n+kfrack+NN+N+N+s N+k+

Solution:
footnotehttps://en.wikipedia.org/wiki/Hockey-stick_identityAbbiamo che _nk^Nnchoose kN+ choose k+. dunque riscrivo il primo termine tramite i coefficienti binomiali: *_n^Nnn+n+n+s n+k_n^Nfracn+k!n-! k+! _n^Nfracn+k!n-!k+! k+! _n^Nn+k choose k+ * e via un index shift otteniamo * _n^Nnn+n+n+s n+kk+! _n^Nn+k choose k+ k+!N+k+ choose k+ frack+ N N+s N+k+ * textbfIn alternativa: con induzione su N e poi su k: * _n^N+&nn+n+n+s n+k _n^Nnn+n+n+s n+k+N+N+N+s N+k+ &oversetPIfrack+NN+N+N+s N+k++N+N+N+s N+k+ leftfracNk++rightN+N+N+s N+k+ leftfracN+k+k+rightN+N+N+s N+k+ frack+ N+N+N+s N+k+ N+k+. * Mentre per induzione completa su k * _n^N+&nn+n+n+s n+k n+k+ _n^N+&nn+n+n+s n+k n+k+ *
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Induzione, Successioni e serie
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(5, default)
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Language
ITA (Italiano)
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Decoration
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