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.

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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.

Show that the Fibonacci numbers satisfy the recurrence relation f_n = 5f_n-4 + 3f_n-5 for n = 5, 6, 7,..., together with the initial f_0 = 0, f_1 = 1, f_2 = 1, f_3 = 2, and f_4 = 3.
See the attachment for the full question.

Using
Fn+3 = Fn+2 + Fn+1
show that
Fn+3 + Fn = 2Fn + 2
and
Fn + 3 - Fn = 2Fn + 1
It seems intuitive at first but would appreciate a detailed explanation - how does one identify the terms to be used?
Thank you

I want to get a better understanding of how these problems are done.
For Exercises #1-3, decide whether the sequences described are subsequences of the Fibonacci sequence, that is, their members are some or all of the members, in the right order, of the Fibonacci sequence.
1. The sequence A(n), where A(n) = (n-1)2^n-2 +

See attachment
1. John wants to fence a 150 square meters rectangular field. He wants the length and width to be natural numbers {1,2,3,...}. What field dimensions will require the least amount of fencing?
2. List the terms that complete a possible pattern in each of the following, and classify each sequence as arit

Suppose that F1 = 1, F2 = 1, F3 =1, F4 = 3, F5 = 5, and in general Fn = Fn-1 + Fn-2 for n ≥ 3 ( Fn is called the nth Fibonacci number.) Prove that F1 + F2 + F3 +...+ Fn = F(n + 2) - 1

Show that |(frac{f_(n+1)}{f_n}) - phi| = frac{1}{(f_n)(phi^{n+1})}
and lim_{n --> infty} frac{f_{n+1}}{f_n} = phi,
where phi is the Golden Ration and is the unique positive root of phi^2 - phi - 1 = 0.
For some discussion on this question, see http://math.stackexchange.com/questions/106049/another-way-to-go-about-provin

First part is to find an expression in terms of n, the results of the formula:
and prove the expression is correct?
Secondly
A Fibonacci sequence is the basis for a superfast calculation trick as follows:
Turn your back and ask someone to write down any two positive integers (vertically and one below the othe

(a) find the first 12 terms of the Fibonacci sequence Fn defined by the Fibonacci relationship
Fn=Fn-1+Fn-2
where F1=1, F2=1.
(b) Show that the ratio of successive F's appears to converge to a number satisfying r2=r+1.
(c) Let r satisfy r2=r+1. Show that the sequence sn=Arn, where A is any constant, satisfies the Fi

Application of Mathematical Induction
It is an application of Mathematical Induction in proving the relations of Fibonacci Numbers.
To prove: F 2n+1 - Fn Fn+2 = (-1) n for n > or, = 3.