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.