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 \(a_1 = 2\), \(a_{n + 1} = a_n/2 + a_n^2/8\) (\(n \geq 1\)).
On the occasion of the beginning of the winter holidays all of the boys from class 8B went to the shooting range. It is known that there are \(n\) boys in 8B. There are \(n\) targets at the shooting range which the class attended. Each of the boys randomly chooses a target, while some of the boys could choose the same target. After this, all of the boys simultaneously attempt to shoot their target. It is known that each of the boys hits their target. The target is considered to be affected if at least one boy has hit it.
a) Find the average number of affected targets.
b) Can the average number of affected targets be less than \(n/2\)?
A fly moves from the origin only to the right or upwards along the lines of the integer grid (a monotonic wander). In each node of the net, the fly randomly selects the direction of further movement: upwards or to the right.
a) Prove that sooner or later the fly will reach the point with abscissa 2011.
b) Find the mathematical expectation of the ordinate of the fly at the moment when the fly reached the abscissa 2011.
Prove that the sequence \(x_n = \sin (n^2)\) does not tend to zero for \(n\) that tends to infinity.