Solutions to this problem are due midnight, Sunday, March 14, 2010

Problem:

We are given a  number a \neq 0.

The sequence x_n is defined by:

x_1=a, x_2=a^2, x_3=a^3

x_n=\frac{x_{n-1}+x_{n-3}}{x_{n-2}} for n\geq 4

How does the value of s(n)=x_n+x_{n-1}+x_{n-2}+x_{n-3}  for large n, depend on a ?

Problem:

The sequence of Fibonacci numbers f_0,f_1 , f_2, \ldots is defined as follows:

f_0=1, f_1=1, f_n=f_{n-1}+f_{n-2} for n\geq 2.

The ratio of successive Fibonacci numbers is r_n=\frac{f_n}{f_{n-1}} .

How fast does the size of the difference \vert r_n-r_{n-1}\vert decay as n increases?

Problem:

For each positive integer n=1, 2, 3, \ldots set E(n)=\frac{1}{n}(n+n^{1/2}+n^{1/3}+\ldots n^{1/n}).

Provide computational evidence that, as n increases without bound, E(n) either:

(a) does not approach a limiting value, or

(b)  does approach a limiting value and say what is that value.

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