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by: Cassidy Grimes
Cassidy Grimes

GPA 3.51

M. Girardi

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M. Girardi
Class Notes
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This 2 page Class Notes was uploaded by Cassidy Grimes on Monday October 26, 2015. The Class Notes belongs to MATH 554 at University of South Carolina - Columbia taught by M. Girardi in Fall. Since its upload, it has received 7 views. For similar materials see /class/229535/math-554-university-of-south-carolina-columbia in Mathematics (M) at University of South Carolina - Columbia.

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Date Created: 10/26/15
PRINCIPLE OF MATHEMATICAL INDUCTION M1 Let N 1 2 3 be the natural numbers Let Z 72710 12 be the integers Let Pn be a statement that is either true or false about 71 Theorem 121 PMI basic form If BASE STEP P1 is true INDUCTIVE STEP for each n e N Pn is true gt Pn 1 is true then Pn is true for each n E N The proof of the PMI is based Peano s Postulates of N Theorem 122 PMI doesn t matter where you start form Fix no 6 Z If BASE STEP Pn0 is true INDUCTIVE STEP for each n E Z with n 2 no Pn is true gt Pn 1 is true then Pn is true for each n E Z such that n 2 no Theorem 123 PMI strong form Fix no 6 Z If BASE STEP Pn0 is true INDUCTIVE STEP for each n e Z with n 2 no Pj istrue forjn01n0n gt Pn1 istrue then Pn is true for each n E Z such that n 2 no Example 1 Prove that n3 lt nl for all integers n 2 6 Example 2 Show that each natural number greater than 3 can be written as a linear comination of the numbers 2 and 5 that is7 for all n 2 4 there exists integers z and y so that n 2x 5y Hint 2m 5y 2 2m 1 5y Reference A An Introduction to Fibonacci Discoueiy by Brother U Alfred The Fibonacci Numbers Fn il The earliest study of these famous numbers is attributed to Leonardo Of Pisa alias Fibonacci early in the thirteenth century They are associated with a variety of natural phenomena They are de ned recursively by Fn Fnil Fnig for 71 Z 3 SOFn 1171727375787m Example 3 A7 page 9 Show that for each n 2 27 F3 7 Fn71Fn1 71 1 Example 4 A7 page 13 Show that for each n E N lt1 5gtn 17 2 2 1 5 17 TT an 2 Fn a Notation suggestion let


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