The Fibonacci numbers are recursively dened by F1 = 1; F2 = 1 and for n...

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The Fibonacci numbers are recursively dened by F1 = 1; F2 = 1and for n > 1; F_(n+1) = F_n + F_(n-1): So the rst few FibonacciNumbers are: 1; 1; 2; 3; 5; 8; 13; 21; 34; 55; 89; 144; : : : Thereare numerous properties of the Fibonacci numbers.
a) Use the principle of Strong Induction to show that allintegers n > 1 and m > 0
F_(n-1)F_(m )+ F_(n)F_(m+1) = F_(n+m):
Solution. (Hint: Use induction with respect to m. First verifythe formula the base case,for m = 1 case (here use again inductionon n) the assume that it is true for m = 1;m = 2; ;m = k then provethat it remains true if m = k + 1 (still with the use of inductionon n)

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3.8 Ratings (682 Votes)

First lets relate the Fibonacci numbers to the following problem Suppose that you want to go up some flight of stairs and at every step you can take either one or two stairs in how many ways can you get up the stairs Well if we say that there are nn stairs then it turns out there are Fn1 ways to do it A very easy inductive proof shows why Base cases If there is 1 stairs See Answer

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