mat3166 list.pdf - MAT 3166 1. Show that for all n ≥ 1 n X...

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MAT 3166Problems + Assignment 11.Show that for alln1,(i)nXk=1k3=n2(n+ 1)24,(ii)nXk=1k(k!) = (n+ 1)!1,(iii)nXk=02knk= 3n.2.Let (an)n0a sequence defined recursively bya0= 0,a1= 1,etan= 3an12an2ifn2.(i) Complutea0, . . . , a5. Conjecture a formula foranand prove it by induction.(ii) LetF=3210. Show that for alln1, we havean+1ananan1=Fn13110.3.Show that for every choice of integersaandbwithb̸= 0, there exist integersqandrwitha=qb+rand−|b|/2< r≤ |b|/2. Is this expression unique?4.Letmandnbe positive integers andrthe remainder of the Euclidian division ofmbyn. Show that the remainder of the division of 2m1 by 2n1 is 2r1.5.Letmandnbe positive integers and letd= gcd(m, n). Using the results of the previousproblem, show that gcd(2m1,2n1) = 2d1.6.In each case, determine the gcd ofa
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Fall
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derioll
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