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Fibonacci Leonard de Pisa 1175 – 1250 AD Best remembered for a problem he posed in Liber Abaci dealing with RABBITS! The Rabbit Problem • At 2 months, the rabbits can reproduce a pair of bunnies. • How many pairs at k months? The Rabbit Problem Months Pairs 1 1 The Rabbit Problem Months Pairs 2 1 The Rabbit Problem Months Pairs 3 2 The Rabbit Problem Months Pairs 4 3 The Rabbit Problem Months Pairs 5 5 The Rabbit Problem Months Pairs 5 8 The next month, all of the gray rabbits that already existed the month before, will make more pairs. The rabbits created this month will not. So we add, B(4)+B(5) = B(6) Recursively defined Fibonacci Sequence Each rabbit from two months ago reproduces a pair. So we add that number to the current number of rabbits. The Fibonacci Sequence Some Properties of the Fibonacci Numbers • No consecutive Fibonacci numbers have a common factor F2 F1 0 F3 F2 2 1 1 F1 F4 F3 3 2 1 F2 F5 F4 5 3 2 F3 No consecutive Fibonacci numbers have a common factor Suppose d is a common factor for Fn and Fn 1 Then d is a factor of Fn 1 Fn Fn 1 Then d is a factor of Fn Fn 1 Fn 2 What can we conclude from this? Some Properties of the Fibonacci Numbers 1. Find the sum of the first five terms of the Fibonacci sequence. 2. Can you find a pattern? Try for six and seven if you can’t see it right away. n F F F i 0 i 1 2 F3 Fn 1 Fn Some Properties of the Fibonacci Numbers F Fn 1 Fn 1 Fn ( Fn 1 Fn 2 ) Fn 1 Fn 1 2 n 1. 2. 3. 4. Distribute the right side, then factor out Fn-1 Simplify Fn Fn 1 Reiterate this process Use this to show F Fn 1 Fn (1) 2 n n 1 for n 2