Problems

The iterative formula of Heron. Prove that the sequence of numbers $\{x_n\}$ given by the conditions $x_1=1$, $x_{n+1}=\frac{1}{2}(x_n+k/x_n)$, 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: $a_0=a>0$, $a_{n+1}=1/3(2a_n+a/a_n^2)$ ($n\ge 0$).

Prove that $\underset{n\to \mathrm{\infty }}{lim}{a}_n=\sqrt[3]{a}$.

The sequence of numbers $\{a_n\}$ is given by $a_1=1$, $a_{n+1}=3a_n/4+1/a_n$ ($n\ge 1$). 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 $a_1=2$, $a_{n+1}=a_n/2+a_n^2/8$ ($n\ge 1$).

The sequence of numbers $\{x_n\}$ is given by the following conditions: $x_1\ge -a$, $x_{n+1}=\sqrt{a+x_n}$. Prove that the sequence $x_n$ is monotonic and bounded. Find its limit.

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 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 $\frac{1}{4}$ 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.