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Pawe ł Janiszewski (146315) Lista 12 Zadanie 2 Tre ść : Zdefiniuj funkcje Fib : obliczaj ą ca n - t ą liczb ę Fibbonacciego jako arytmetyczna funkcje rekurencyjn ą : N N 1. bez u ż ycia rekursji prostej, 2. z u ż yciem rekursji prostej, 3. z u ż yciem rekursji prostej jednoczesnej. Rozwi ą zanie: Ad 1. n i i i n i a a a a a a n a n Fib )) ) ( ) ( ) )(( ( 1 ) ( 1 ) ( 1 ) (ln( (min ) ( 2 1 3 2 1 + = = = + = = Spo ś ród ci ą gów liczb naturalnych d ł ugo ś ci n+1 wybieramy ten którego kolejne elementy spe ł niaj ą dane zale ż no ś ci ci ą gu Fibbonacciego. W tym ci ą gu ostatnim elementem (n-ty poniewa ż na pierwszym miejscu zapisana jest d ł ugo ść tego ci ą gu) jest interesuj ą ca nas n - ta liczba Fibbonacciego. Przyk ł adowy ci ą g : n F n ,..., 8 , 5 , 3 , 2 , 1 , 1 , 1 + Ad 2. ) ( ), ( ) ( ) 0 , ( 0 1 x const x const x g x f = = 0 1 0 )) , ( ( , )) , ( ( )) , ( ( ) , ), , ( ( ) 1 , ( n x f n x f n x f n x n x f h n x f + = = + 0 )) , ( ( ) ( n x f n Fib = Dowód indukcyjny. 1. Dla n = 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 0 , 1 ) ( ), ( ) ( ) ( ), ( , ) ( ), ( ) ( ), ( )) 0 , ( ( , )) 0 , ( ( )) 0 , ( ( )) , ), 0 , ( ( ( )) 1 , ( ( ) 1 ( 0 0 0 0 1 0 0 0 1 1 0 1 0 0 1 0 0 1 0 1 0 = = + = + = + = = = x const x const x const x const x

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