Problems

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Found: 18

The Babylonian algorithm for deducing 2. The sequence of numbers {xn} is given by the following conditions: x1=1, xn+1=12(xn+2/xn) (n1).

Prove that limnxn=2.

The iterative formula of Heron. Prove that the sequence of numbers {xn} given by the conditions x1=1, xn+1=12(xn+k/xn), converges. Find the limit of this sequence.

The algorithm of the approximate calculation of a3. The sequence {an} is defined by the following conditions: a0=a>0, an+1=1/3(2an+a/an2) (n0).

Prove that limnan=a3.

The sequence of numbers {an} is given by a1=1, an+1=3an/4+1/an (n1). Prove that:

a) the sequence {an} converges;

b) |a10002|<(3/4)1000.

The sequence of numbers {xn} is given by the following conditions: x1a, xn+1=a+xn. Prove that the sequence xn is monotonic and bounded. Find its limit.

We call the geometric-harmonic mean of numbers a and b the general limit of the sequences {an} and {bn} constructed according to the rule a0=a, b0=b, an+1=2anbnan+bn, bn+1=anbn (n0).

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)×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).

Definition. The sequence of numbers a0,a1,,an,, which, with the given p and q, satisfies the relation an+2=pan+1+qan (n=0,1,2,) is called a linear recurrent sequence of the second order.

The equation x2pxq=0 is called a characteristic equation of the sequence {an}.

Prove that, if the numbers a0, a1 are fixed, then all of the other terms of the sequence {an} are uniquely determined.

The frog jumps over the vertices of the hexagon ABCDEF, 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?