We solve problems

Problem #PRU-109715

Problems Algebra and arithmetic Sequences Progressions Geometric progression Calculus Real numbers Integer and fractional parts. Archimedean property Sums of number sequences and difference series

13–15 4.0

Find the sum $1 / 3 + 2 / 3 + 2^{2} / 3 + 2^{3} / 3 + \dots + 2^{1000} / 3$ .

Problem #PRU-61299

Problems Calculus Number sequences Limit of a sequence, convergence Algebra and arithmetic 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 and arithmetic 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 and arithmetic 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 and arithmetic 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 and arithmetic 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-100312

Problems Algebra and arithmetic Sequences Progressions Arithmetic progression

There are one hundred natural numbers, they are all different, and sum up to 5050. Can you find those numbers? Are they unique, or is there another bunch of such numbers?

Problem #PRU-61474

Problems Probability and statistics Probability theory Discrete distribution Algebra and arithmetic Sequences Recurrent relations Linear recurrent relations

15–16 4.0

The frog jumps over the vertices of the hexagon $A B C D E F$ , 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?

Problem #PRU-65275

Problems Probability and statistics Probability theory Discrete distribution Algebra and arithmetic Sequences Recurrent relations Fibonacci numbers

14–16 4.0

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).

Problem #PRU-65308

Problems Probability and statistics Probability theory Discrete distribution Algebra and arithmetic 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).