We call the geometric-harmonic mean of numbers \(a\) and \(b\) the general limit of the sequences \(\{a_n\}\) and \(\{b_n\}\) constructed according to the rule \(a_0 = a\), \(b_0 = b\), \(a_{n + 1} = \frac{2a_nb_n}{a_n + b_n}\), \(b_{n + 1} = \sqrt{a_nb_n}\) (\(n \geq 0\)).
We denote it by \(\nu (a, b)\). Prove that \(\nu (a, b)\) is related to \(\mu (a, b)\) (see problem number 61322) by \(\nu (a, b) \times \mu (1/a, 1/b) = 1\).
Problem number 61322 says that both of these sequences have the same limit.
This limit is called the arithmetic-geometric mean of the numbers \(a, b\) and is denoted by \(\mu (a, b)\).
Let the sequences of numbers \(\{a_n\}\) and \(\{b_n\}\), that are associated with the relation \(\Delta b_n = a_n\) (\(n = 1, 2, \dots\)), be given. How are the partial sums \(S_n\) of the sequence \(\{a_n\}\) \(S_n = a_1 + a_2 + \dots + a_n\) linked to the sequence \(\{b_n\}\)?
Definition. The sequence of numbers \(a_0, a_1, \dots , a_n, \dots\), which, with the given \(p\) and \(q\), satisfies the relation \(a_{n + 2} = pa_{n + 1} + qa_n\) (\(n = 0,1,2, \dots\)) 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?
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).
\(A\) and \(B\) shoot in a shooting gallery, but they only have one six-shot revolver with one cartridge. Therefore, they agreed in turn to randomly rotate the drum and shoot. \(A\) goes first. Find the probability that the shot will occur when \(A\) has the revolver.
A coin is thrown 10 times. Find the probability that it never lands on two heads in a row.
In the magical land of Anchuria there is a drafts championship made up of several rounds. The days and cities in which the rounds are carried out are determined by a draw. According to the rules of the championship, no two rounds can take place in one city, and no two rounds can take place on one day. Among the fans, a lottery is arranged: the main prize is given to those who correctly guess, before the start of the championship, in which cities and on which days all of the round will take place. If no one guesses, then the main prize will go to the organising committee of the championship. In total, there are eight cities in Anchuria, and the championship is only allotted eight days. How many rounds should there be in the championship, so that the organising committee is most likely to receive the main prize?
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 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).