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UMD CMSC 250 - Chapter 4 Mathematical Induction

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1Chapter 4Mathematical Induction• Used to verify a property of a sequence• 2,4,6,8,… for i >= 1 ai= 2i– infinite sequence with infinite distinct values• for i >= 1 bi= (-1)i– infinite sequence with finite distinct values• for 1<=i<=6 ci= i+5– finite sequence (with finite distinct values)Finding the Explicit Formula• Figure the formula of this sequence• Different sequences with same initial values,...251,161,91,41,1−−2)1(1203++−=+=>=kkbkakkk2Summation & Product Notation• Sum of Items Specified• Product of Items Specified654321612222222 +++++==kk)5(2*)4(2*)3(2*)2(2*)1(2251=Π=kkVariable ending point • n as the index of the final term• for n = 2• for n = 3=++nkknk013Nesting of Sum/Product Notation• Variations (same or different??): = =JjniijjY121)( = =JjniijjY121)( = =JjniijjY1 12Telescoping Series=++−+nkkkkk1)211()1(1+Π=iini4Properties• Merging and Splitting• Distribution===+=+nmkkknmkknmkkbaba )()(** kknmkknmkknmkbabaΠΠΠ=======nmkknmkkacac )*(*+===+=nikkimkknmkkaaa1knikkimkknmkaaaΠΠΠ+====1*Factorial• n! = n*(n-1)*(n-2)*…*2*1• Definition0! = 1n! =


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UMD CMSC 250 - Chapter 4 Mathematical Induction

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