(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 Fibonacci relationship. Is the Fibonnaci sequence that you found in (a) given by this formula for some A?

(d) Notice that the quadratic equation satisfied by r above has two roots. Let them be r1 and r2, and show that Ar1n+Br2n satisfies the Fibonnaci relationship, for any choice of constants A and B.

(e) Use the observation in (d) to find a formula for the nth term of the Fibonnaci sequence, and prove that it works.

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Fibonacci Sequences, Convergence and Limits are investigated. The solution is detailed and well presented.

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

The sequence Sn = ((1+ (1/n))^n converges, and its limit can be used to define e.
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Xn(n+1-nx) < 1
b) Substitute the following x-value into the inequality from part (a)

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.
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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?
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and prove the expression is correct?
Secondly
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In each case, briefly explain your answer including justification for the value of the limit (if it exists)
a) (1/3)ⁿ
b