Induction for recursively defined sequences

For $f$ a real valued function on the real line, define the function $\Delta f$ by $\Delta f(x) = f(x+1)-f(x)$. For $n \geq 2$, define $\Delta^{n} f$ recursively by $\Delta^{n} f = \Delta (\Delta^{n-1} f)$. Prove that $\Delta^{n} f = 0$ if and only if $f$ has the form $f(x) = a_{0}(x)+a_{1}(x)x+\cdots +a_{n-1}(x)x^{n-1}$ where $a_{0}, a_{1}, \ldots , a_{n-1}$ are periodic functions of period 1.

Prove that if $x_{1} = a^{1 / k}(a > 0), x_{n+1} = \left(a / x_{n}\right)^{1 / k}$, $n \in \mathbb {N}$, then $x_{n} = a^{\left(1-(-k)^{-n}\right) /(k+1)}$.

Let

where $a, \alpha , \beta$ are positive numbers. Find a formula for the general term of this sequence and the index of its largest term.

Let $u(n)$ be the sequence of Fibonacci numbers. Prove that:

Let $\quad u_{0}(t)=0, \quad u_{1}(t)=1 \quad$ and $\quad u_{n}(t)=t u_{n-1}(t)-u_{n-2}(t)$. Prove that:

The Fibonacci numbers $F_{i}, i \in \omega$, are defined as follows: $F_{i}=i$ for $i<2$, $F_{i}=F_{i-1}+F_{i-2}$ otherwise. Prove that $F_{i}$ is even if and only if $i$ is divisible by 3.

Prove the identity $F_{i-1} F_{i+1}=F_{i}^{2}+(-1)^{i}$ for the Fibonacci numbers, for $i>0$.

The Fibonacci numbers are defined by the following recurrence relations: $F_{0}=0, F_{1}=1, F_{n+2}=F_{n+1}+F_{n}$. Prove that

Prove that for $n \geqslant 1$

The Tower of Hanoi problem. There are three pegs. Several disks of decreasing diameter from bottom to top are placed on one of them (forming a children's pyramid toy). It is allowed to move the top disk from any peg to any other, but it is forbidden to place a disk of larger diameter on top of a disk of smaller diameter. The task is to move the entire pyramid to another peg.

The g.r.f. (general recursive function) $f(x)$ is given as follows:

Prove that $f(x)=3 \cdot 2^{x+1}-4 x-6$. $\circledast$