We solve problems

Problem #PRU-109599

Problems Calculus Number sequences Boundedness, monotonicity Algebra Polynomials Algebraic identities for polynomials Identical transformations Sequences

14–16 5.0

Prove that for any natural number $a_{1} > 1$ there exists an increasing sequence of natural numbers $a_{1}, a_{2}, a_{3}, \dots$ , for which $a_{1}^{2} + a_{2}^{2} + \dots + a_{k}^{2}$ is divisible by $a_{1} + a_{2} + \dots + a_{k}$ for all $k \geq 1$ .

Problem #PRU-109603

Problems Algebra and arithmetic Number theory. Divisibility Divisibility of a number. General properties Methods Examples and counterexamples. Constructive proofs Calculus Number sequences Number sequences (other)

14–16 5.0

Is there a sequence of natural numbers in which every natural number occurs exactly once, and for any $k = 1, 2, 3, \dots$ the sum of the first $k$ terms of the sequence is divisible by $k$ ?

Problem #PRU-60420

Problems Calculus Number sequences Number sequences (other)

15–16 3.0

Which term in the expansion $(1 + \sqrt{3})^{100}$ will be the largest by the Newton binomial formula?

Problem #PRU-60846

Problems Algebra and arithmetic Algebraic inequalities and systems of inequalities Linear inequalites and systems of inequalities Calculus Number sequences Number sequences (other)

15–16 3.0

Prove that in any infinite decimal fraction you can rearrange the numbers so that the resulting fraction becomes a rational number.

Problem #PRU-61297

Problems Calculus Number sequences Limit of a sequence, convergence Algebra and arithmetic 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 and arithmetic 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 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-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-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) $| a_{1000} - 2 | < (3 / 4)^{1000}$ .