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# Fibonacci Sequences

Fibonacci posed the following problem: Suppose that rabbits live forever and that every month each pair produces a new pair which becomes productive at age 2 months. If we start with one newborn pair, how many pairs of rabbits will we have in the nth month? Show that the answer is fn, where {fn} is the Fibonacci sequence defined in Example 2(c)

From the example 2(c): The Fibonacci sequence {fn} is defined recursively by the conditions
f1 = 1, f2 = 1, fn = fn-1 + fn-2 where n is greater or equal to 3

Each term is the sum of the two preceding terms. The first few terms are {1,1,2,3,5,8,13,21,...}

b) Let an = fn+1/fn and show that an-1 = 1+1/an-2 assuming that {an} is convergent, find its limit.

#### Solution Preview

Fibonacci posed the following problem: Suppose that rabbits live forever and that every month each pair produces a new pair which becomes productive at age 2 months. If we start with one newborn pair, how many pairs of rabbits will we have in the nth month? Show that the answer is fn, where {fn} is the Fibonacci sequence defined in Example 2(c)

From the example 2(c): The Fibonacci sequence {fn} is defined recursively by the conditions
f1 = 1, f2 = 1, fn = fn-1 + fn-2 where n is greater or equal to 3

Each term is the sum of ...

#### Solution Summary

Fibonacci Sequences are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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