We solve problems

Problem #PRU-61297

Problems Calculus Number sequences Limit of a sequence, convergence Algebra Sequences Recurrent relations Recurrent relations (other)

15–16 3.0

The Babylonian algorithm for deducing $\sqrt{2}$ . The sequence of numbers ${x_{n}}$ is given by the following conditions: $x_{1} = 1$ , $x_{n + 1} = \frac{1}{2} (x_{n} + 2 / x_{n})$ ( $n \geq 1$ ).

Prove that $lim_{n \to \infty} x_{n} = \sqrt{2}$ .

Problem #PRU-61298

Problems Calculus Number sequences Limit of a sequence, convergence Algebra Sequences Recurrent relations Recurrent relations (other)

15–16 3.0

What will the sequence from the previous problem 61297 be converging towards if we choose $x_{1}$ as equal to $- 1$ as the initial condition?

Problem #PRU-61299

Problems Calculus Number sequences Limit of a sequence, convergence Algebra Sequences Recurrent relations Recurrent relations (other)

15–16 4.0

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.

Problem #PRU-61304

Problems Calculus Functions of one variable. Continuity Continuous functions (general properties) Number sequences Limit of a sequence, convergence

15–16 3.0

Method of iterations. In order to approximately solve an equation, it is allowed to write $f (x) = x$ , by using the iteration method. First, some number $x_{0}$ is chosen, and then the sequence ${x_{n}}$ is constructed according to the rule $x_{n + 1} = f (x_{n})$ ( $n \geq 0$ ). Prove that if this sequence has the limit $x * = lim_{n \to \infty} x_{n}$ , and the function $f (x)$ is continuous, then this limit is the root of the original equation: $f (x^{*}) = x^{*}$ .

Problem #PRU-61308

Problems Calculus Number sequences Limit of a sequence, convergence

15–16 3.0

The numbers $a_{1}, a_{2}, \dots, a_{k}$ are such that the equality $lim_{n \to \infty} (x_{n} + a_{1} x_{n - 1} + \dots + a_{k} x_{n - k}) = 0$ is possible only for those sequences ${x_{n}}$ for which $lim_{n \to \infty} x_{n} = 0$ . Prove that all the roots of the polynomial P $(λ) = λ^{k} + a_{1} λ^{k - 1} + a_{2} λ^{k - 2} + \dots + a_{k}$ are modulo less than 1.

Problem #PRU-61311

Problems Calculus Number sequences Limit of a sequence, convergence Algebra Sequences Recurrent relations Recurrent relations (other)

15–16 4.0

The algorithm of the approximate calculation of $\sqrt[3]{a}$ . The sequence ${a_{n}}$ is defined by the following conditions: $a_{0} = a > 0$ , $a_{n + 1} = 1 / 3 (2 a_{n} + a / a_{n}^{2})$ ( $n \geq 0$ ).

Prove that $lim_{n \to \infty} a_{n} = \sqrt[3]{a}$ .

Problem #PRU-61313

Problems Calculus Number sequences Limit of a sequence, convergence Algebra Sequences Recurrent relations Recurrent relations (other)

15–16 4.0

The sequence of numbers ${a_{n}}$ is given by $a_{1} = 1$ , $a_{n + 1} = 3 a_{n} / 4 + 1 / a_{n}$ ( $n \geq 1$ ). Prove that:

a) the sequence ${a_{n}}$ converges;

b) $| a_{1000} - 2 | < (3 / 4)^{1000}$ .

Problem #PRU-61314

Problems Calculus Number sequences Limit of a sequence, convergence Algebra Sequences Recurrent relations Recurrent relations (other)

15–16 4.0

Find the limit of the sequence that is given by the following conditions $a_{1} = 2$ , $a_{n + 1} = a_{n} / 2 + a_{n}^{2} / 8$ ( $n \geq 1$ ).

Problem #PRU-61324

Problems Calculus Number sequences Limit of a sequence, convergence Algebra Mean values Sequences Recurrent relations Recurrent relations (other)

15–16 3.0

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 $a_{0} = a$ , $b_{0} = b$ , $a_{n + 1} = \frac{2 a_{n} b_{n}}{a_{n} + b_{n}}$ , $b_{n + 1} = \sqrt{a_{n} b_{n}}$ ( $n \geq 0$ ).

We denote it by $ν (a, b)$ . Prove that $ν (a, b)$ is related to $μ (a, b)$ (see problem number 61322) by $ν (a, b) \times μ (1 / a, 1 / b) = 1$ .

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 $μ (a, b)$ .

Problem #PRU-65276

Problems Probability and statistics Probability theory Discrete distribution Calculus Number sequences Limit of a sequence, convergence Number e

14–16 4.0

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 $n / 2$ ?