We solve problems

Problem #PRU-61306

Problems Calculus Number sequences Boundedness, monotonicity Algebra and arithmetic Sequences Recurrent relations Recurrent relations (other)

13–15 3.0

The sequence of numbers $a_{n}$ is given by the conditions $a_{1} = 1$ , $a_{n + 1} = a_{n} + 1 / a_{n}^{2}$ ( $n \geq 1$ ).

Is it true that this sequence is limited?

Problem #PRU-61458

Problems Algebra and arithmetic Sequences Recurrent relations Linear recurrent relations Methods Mathematical induction

13–16 2.0

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 $x^{2} - p x - q = 0$ 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.

Problem #PRU-65275

Problems Probability and statistics Probability theory Discrete distribution Algebra and arithmetic Sequences Recurrent relations Fibonacci numbers

14–16 4.0

The figure shows the scheme of a go-karting route. The start and finish are at point $A$ , and the driver can go along the route as many times as he wants by going to point $A$ and then back onto the circle.

It takes Fred one minute to get from $A$ to $B$ or from $B$ to $A$ . It also takes one minute for Fred to go around the ring and he can travel along the ring in an anti-clockwise direction (the arrows in the image indicate the possible direction of movement). Fred does not turn back halfway along the route nor does not stop. He is allowed to be on the track for 10 minutes. Find the number of possible different routes (i.e. sequences of possible routes).

Problem #PRU-65289

Problems Probability and statistics Probability theory Discrete distribution Algebra and arithmetic Sequences Recurrent relations Fibonacci numbers

13–16 3.0

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

Problem #PRU-65296

Problems Probability and statistics Probability theory Conditional probability Algebra and arithmetic Algebraic inequalities and systems of inequalities Linear inequalites and systems of inequalities Sequences Recurrent relations Linear recurrent relations

13–16 3.0

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.

Problem #PRU-65308

Problems Probability and statistics Probability theory Discrete distribution Algebra and arithmetic Mean values Sequences Recurrent relations Recurrent relations (other)

14–16 4.0

A regular dice is thrown many times. Find the mathematical expectation of the number of rolls made before the moment when the sum of all rolled points reaches 2010 (that is, it became no less than 2010).

Problem #PRU-65311

Problems Probability and statistics Probability theory Conditional probability Discrete distribution Algebra and arithmetic Mean values Sequences Recurrent relations Recurrent relations (other)

14–16 4.0

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.

Problem #PRU-65777

Problems Probability and statistics Probability theory Conditional probability Discrete distribution Algebra and arithmetic Algebraic inequalities and systems of inequalities Linear inequalites and systems of inequalities Sequences Recurrent relations Linear recurrent relations

14–16 4.0

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.

Problem #PRU-66056

Problems Probability and statistics Probability theory Discrete distribution Set theory and logic Set theory Inclusion-exclusion principle Algebra and arithmetic Sequences Recurrent relations Recurrent relations (other)

14–16 4.0

An incredible legend says that one day Stirling was considering the numbers of Stirling of the second kind. During his thoughtfulness, he threw 10 regular dice on the table. After the next throw, he suddenly noticed that in the dropped combination of points there were all of the numbers from 1 to 6. Immediately Stirling reflected: what is the probability of such an event? What is the probability that when throwing 10 dice each number of points from 1 to 6 will drop out on at least one die?

Problem #PRU-98427

Problems Algebra and arithmetic Sequences Recurrent relations Linear recurrent relations Calculus Functions of one variable. Continuity Periodicity and aperiodicity

12–14 3.0

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.