Školjka

%V0 Let $\{fn\}$ be the Fibonacci sequence $\{1, 1, 2, 3, 5, \dots.\}.$ (a) Find all pairs $(a, b)$ of real numbers such that for each $n$, $af_n +bf_{n+1}$ is a member of the sequence. (b) Find all pairs $(u, v)$ of positive real numbers such that for each $n$, $uf_n^2 +vf_{n+1}^2$ is a member of the sequence.

%V0 A sequence of real numbers $u_1, u_2, u_3, \dots$ is determined by $u_1$ and the following recurrence relation for $n \geq 1$: $$4u_{n+1} = \sqrt[3]{ 64u_n + 15.}$$ Describe, with proof, the behavior of $u_n$ as $n \to \infty.$

%V0 Define sequence $(a_n)$ by $\sum_{d|n} a_d = 2^n.$ Show that $n|a_n.$

%V0 Let $p(x)$ be a cubic polynomial with rational coefficients. $q_1$, $q_2$, $q_3$, ... is a sequence of rationals such that $q_n = p(q_{n + 1})$ for all positive $n$. Show that for some $k$, we have $q_{n + k} = q_n$ for all positive $n$.

%V0 Let $a_n$ be the last nonzero digit in the decimal representation of the number $n!.$ Does the sequence $a_1, a_2, \ldots, a_n, \ldots$ become periodic after a finite number of terms?

%V0 Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x.$ Pick any $x_1$ in $[0, 1)$ and define the sequence $x_1, x_2, x_3, \ldots$ by $x_{n+1} = 0$ if $x_n = 0$ and $x_{n+1} = \frac{1}{x_n} - \left \lfloor \frac{1}{x_n} \right \rfloor$ otherwise. Prove that $$x_1 + x_2 + \ldots + x_n < \frac{F_1}{F_2} + \frac{F_2}{F_3} + \ldots + \frac{F_n}{F_{n+1}},$$ where $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1} + F_n$ for $n \geq 1.$