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#### Linear recurrent relations , Periodicity and aperiodicity

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.

#### Sequences

For which natural $n$ does the number $\frac{n^2}{1,001^n}$ reach its maximum value?

#### Boundedness, monotonicity , Sequnces

$a_1$, $a_2$, $a_3$, … is an increasing sequence of natural numbers. It is known that $a_{a_k} = 3k$ for any $k.$ Find a$)$ $a_{100}$; b$)$ $a_{2022}$.

#### Arithmetic progression

Is the sum of the numbers 1 + 2 + 3 + … + 1999 divisible by 1999?

#### Conditional probability , Discrete distribution , Mean values , Recurrent relations (other)

A fair dice is thrown many times. It is known that at some point the total amount of points became equal to exactly 2010.
Find the mathematical expectation of the number of throws made to this point.

#### Discrete distribution , Fibonacci numbers

A coin is thrown 10 times. Find the probability that it never lands on two heads in a row.

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