The iterative formula of Heron. Prove that the sequence of numbers \(\{x_n\}\) given by the conditions \(x_1 = 1\), \(x_{n + 1} = \frac 12 (x_n + k/x_n)\), converges. Find the limit of this sequence.
The algorithm of the approximate calculation of \(\sqrt[3]{a}\). The sequence \(\{a_n\}\) is defined by the following conditions: \(a_0 = a > 0\), \(a_{n + 1} = 1/3 (2a_n + a/a^2_n)\) (\(n \geq 0\)).
Prove that \(\lim\limits_{n\to\infty} a_n = \sqrt[3]{a}\).
The sequence of numbers \(\{a_n\}\) is given by \(a_1 = 1\), \(a_{n + 1} = 3a_n/4 + 1/a_n\) (\(n \geq 1\)). Prove that:
a) the sequence \(\{a_n\}\) converges;
b) \(|a_{1000} - 2| < (3/4)^{1000}\).
Find the limit of the sequence that is given by the following conditions \(a_1 = 2\), \(a_{n + 1} = a_n/2 + a_n^2/8\) (\(n \geq 1\)).
The sequence of numbers \(\{x_n\}\) is given by the following conditions: \(x_1 \geq - a\), \(x_{n + 1} = \sqrt{a + x_n}\). Prove that the sequence \(x_n\) is monotonic and bounded. Find its limit.
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).
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.
Hercules meets the three-headed snake, the Lernaean Hydra and the battle begins. Every minute, Hercules cuts one of the snake’s heads off. With probability \(\frac 14\) in the place of the chopped off head grows two new ones, with a probability of \(1/3\), only one new head will grow and with a probability of \(5/12\), not a single head will appear. The serpent is considered defeated if he does not have a single head left. Find the probability that sooner or later Hercules will beat the snake.
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?