Problems

The sequence of numbers $a_1,a_2,\dots$ is given by the conditions $a_1=1$, $a_2=143$ and

for all $n\ge 2$.

Prove that all members of the sequence are integers.

Definition. The sequence of numbers $a_0,a_1,\dots ,a_n,\dots$, which, with the given $p$ and $q$, satisfies the relation $a_{n+2}=p{a}_{n+1}+q{a}_n$ ($n=0,1,2,\dots$) is called a linear recurrent sequence of the second order.

The equation $$x^2-px-q=0$$ is called a characteristic equation of the sequence $\{a_n\}$.

Prove that, if the numbers $a_0$, $a_1$ are fixed, then all of the other terms of the sequence $\{a_n\}$ are uniquely determined.

The frog jumps over the vertices of the hexagon $ABCDEF$, each time moving to one of the neighbouring vertices.

a) How many ways can it get from $A$ to $C$ in $n$ jumps?

b) The same question, but on condition that it cannot jump to $D$?

c) Let the frog’s path begin at the vertex $A$, and at the vertex $D$ there is a mine. Every second it makes another jump. What is the probability that it will still be alive in $n$ seconds?

d)* What is the average life expectancy of such frogs?

On a calculator keypad, there are the numbers from 0 to 9 and signs of two actions (see the figure). First, the display shows the number 0. You can press any keys. The calculator performs the actions in the sequence of clicks. If the action sign is pressed several times, the calculator will only remember the last click.

a) The button with the multiplier sign breaks and does not work. The Scattered Scientist pressed several buttons in a random sequence. Which result of the resulting sequence of actions is more likely: an even number or an odd number?

b) Solve the previous problem if the multiplication symbol button is repaired.

A high rectangle of width 2 is open from above, and the L-shaped domino falls inside it in a random way (see the figure).

a) $k$ $L$-shaped dominoes have fallen. Find the mathematical expectation of the height of the resulting polygon.

b) $7$ $G$-shaped dominoes fell inside the rectangle. Find the probability that the resulting figure will have a height of 12.

In a row there are 2023 numbers. The first number is 1. It is known that each number, except the first and the last, is equal to the sum of two neighboring ones. Find the last number.