Math 55 WorksheetAdapted from worksheets by Rob Bayer, Summer 2009.Induction1. Prove that the sum of the firstnodd numbers isn2.2. Using the product ruleddx(f(x)g(x)) =f0(x)g(x) +f(x)g0(x), and the fact thatddxx= 1, use induction to provethatddx(xn) =nxn-1.3. Letanbe the sequence defined bya1=√2,an=√2an-1.(a) Show that 1< an<2 for alln.(b) Show thatan+1> anfor alln.NOTE: A sequence like this is called bounded and monotone and is guaranteed to converge.4. The harmonic numbers are defined byHn= 1 +12+13+14+· · ·+1n.(a) Show thatH2n≥1 +n2for alln.(b) Use your answer to (a) to show that∞Xn=11nincreases without bound.5. Prove that 1 + 23+ 33+ 43+· · ·+n3=n(n+ 1)22.Strong Induction and Well-ordering1. Prove that ifn≥18, then you can makencents out of just 4- and 7-cent stamps2. Here we’ll use well-ordering to show thatx2+y2= 3xyzhas no solutions in positive integers.
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