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-60468

Problems Algebra and arithmetic Sequences Progressions Arithmetic progression Calculus Functions of one variable. Continuity Certain properties of a function and recurrence relations.

13–15 3.0

Suppose that there are 15 prime numbers forming an arithmetic progression with a difference of $d$ . Prove that $d > 30, 000$ .

Problem #PRU-97976

Problems Algebra and arithmetic Sequences

14–16 5.0

We consider a sequence of words consisting of the letters “A” and “B”. The first word in the sequence is “A”, the $k$ -th word is obtained from the $(k - 1)$ -th by the following operation: each “A” is replaced by “AAB” and each “B” by “A”. It is easy to see that each word is the beginning of the next, thus obtaining an infinite sequence of letters: AABAABAAABAABAAAB...

a) Where in this sequence will the 1000th letter “A” be?

b) Prove that this sequence is non-periodic.

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-60364

Problems Algebra and arithmetic Sequences Progressions Arithmetic progression Methods Pigeonhole principle Pigeonhole principle (other) Word problems Tables and tournaments Tournament tables

12–14 3.0

In a volleyball tournament teams play each other once. A win gives the team 1 point, a loss 0 points. It is known that at one point in the tournament all of the teams had different numbers of points. How many points did the team in second last place have at the end of the tournament, and what was the result of its match against the eventually winning team?

Problem #PRU-61306

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

13–15 3.0

The sequence of numbers $a_{n}$ is given by the conditions $a_{1} = 1$ , $a_{n + 1} = a_{n} + 1 / a_{n}^{2}$ ( $n \geq 1$ ).

Is it true that this sequence is limited?

Problem #PRU-61429

Problems Algebra and arithmetic Sequences Sums of number sequences and difference series

13–16 2.0

Let the sequences of numbers ${a_{n}}$ and ${b_{n}}$ , that are associated with the relation $Δ b_{n} = a_{n}$ ( $n = 1, 2, \dots$ ), be given. How are the partial sums $S_{n}$ of the sequence ${a_{n}}$ $S_{n} = a_{1} + a_{2} + \dots + a_{n}$ linked to the sequence ${b_{n}}$ ?

Problem #PRU-61458

Problems Algebra and arithmetic Sequences Recurrent relations Linear recurrent relations Methods Mathematical induction

13–16 2.0

Definition. The sequence of numbers $a_{0}, a_{1}, \dots, a_{n}, \dots$ , which, with the given $p$ and $q$ , satisfies the relation $a_{n + 2} = p a_{n + 1} + q a_{n}$ ( $n = 0, 1, 2, \dots$ ) is called a linear recurrent sequence of the second order.

The equation $x^{2} - p x - q = 0$ is called a characteristic equation of the sequence ${a_{n}}$ .

Prove that, if the numbers $a_{0}$ , $a_{1}$ are fixed, then all of the other terms of the sequence ${a_{n}}$ are uniquely determined.

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-65277

Problems Probability and statistics Probability theory Discrete distribution Algebra and arithmetic Sequences Progressions Geometric progression

13–16 3.0

$A$ and $B$ shoot in a shooting gallery, but they only have one six-shot revolver with one cartridge. Therefore, they agreed in turn to randomly rotate the drum and shoot. $A$ goes first. Find the probability that the shot will occur when $A$ has the revolver.