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-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-61319

Problems Calculus Number sequences Boundedness, monotonicity Algebra Sequences Recurrent relations Recurrent relations (other)

15–16 4.0

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.

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-65308

Problems Probability and statistics Probability theory Discrete distribution Algebra Mean values Sequences Recurrent relations Recurrent relations (other)

14–16 4.0

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).

Problem #PRU-65311

Problems Probability and statistics Probability theory Conditional probability Discrete distribution Algebra Mean values Sequences Recurrent relations Recurrent relations (other)

14–16 4.0

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.