In 1202, Leonardo Fibonacci posed the following problem: Suppose a particular breed of rabbit breeds one new pair of rabbits each month, except that a 1-month-old pair is too young to breed. Suppose further that no rabbit breeds with any other except its paired mate and that rabbits live forever. At 1 month we have our original pair of rabbits. At 2 months we still have the single pair. At 3 months, we have two pairs (the original and their one pair of offspring). At 4 months we have three pairs (the original pair, one older pair of offspring, and one new pair of offspring).
(a) Show that at n months, there are fn pairs of rabbits.
(b) Calculate the first ten Fibonacci numbers f1, f2, f3, Á , f10.
(c) Find a formula for fn+3 − fn+1.
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