Definition. The sequence of numbers \(a_0, a_1, \dots , a_n, \dots\), which, with the given \(p\) and \(q\), satisfies the relation \(a_{n + 2} = pa_{n + 1} + qa_n\) (\(n = 0,1,2, \dots\)) is called a linear recurrent sequence of the second order.
The equation \[x^2-px-q = 0\] 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.
A frog jumps over the vertices of the triangle \(ABC\), moving each time to one of the neighbouring vertices.
How many ways can it get from \(A\) to \(A\) in \(n\) jumps?
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?
Let \((1 + \sqrt {2} + \sqrt {3})^n = p_n + q_n \sqrt {2} + r_n \sqrt {3} + s_n \sqrt {6}\) for \(n \geq 0\). Find:
a) \(\lim \limits_ {n \to \infty} {\frac {p_n} {q_n}}\); b) \(\lim \limits_ {n \to \infty} {\frac {p_n} {r_n}}\); c) \(\lim \limits_ {n \to \infty} {\frac {p_n} {s_n}}\);
Find the generating functions of the sequences of Chebyshev polynomials of the first and second kind: \[F_T(x,z) = \sum_{n=0}^{\infty}T_n(x)z^n;\quad F_U(x,z) = \sum_{n=0}^{\infty}U_n(X)z^n.\]
Definitions of Chebyshev polynomials can be found in the handbook.
We denote by \(P_{k, l}(n)\) the number of partitions of the number \(n\) into at most \(k\) terms, each of which does not exceed \(l\). Prove the equalities:
a) \(P_{k, l}(n) - P_{k, l-1}(n) = P_{k-1, l}(n-l)\);
b) \(P_{k, l}(n) - P_{k-1, l} (n) = P_{k, l-1}(n-k)\);
c) \(P_{k, l}(n) = P_{l, k} (n)\);
d) \(P_{k, l}(n) = P_{k, l} (kl - n)\).
Prove that the 13th day of the month is more likely to occur on a Friday than on other days of the week. It is assumed that we live in the Gregorian style calendar.
Find the coefficient of \(x\) for the polynomial \((x - a) (x - b) (x - c) \dots (x - z)\).
The following words/sounds are given: look, yar, yell, lean, lease. Determine what will happen if the sounds that make up these words are pronounced in reverse order.
Author: D.E. Shnol
On the island of Truthland, all of the inhabitants may be mistaken, but the younger ones never contradict the elders, and when the older ones contradict the younger ones, they (the elders) are not mistaken. Between the residents A, B and C there was such a conversation:
A: B is the tallest.
B: A is the tallest.
C: I’m taller than B.
Does it follow from this conversation that the younger the person, the taller he or she is (for the three people having this conversation)?