Let the sequences of numbers \(\{a_n\}\) and \(\{b_n\}\), that are associated with the relation \(\Delta b_n = a_n\) (\(n = 1, 2, \dots\)), 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 \(\{b_n\}\)?
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?
Find the coefficient of \(x\) for the polynomial \((x - a) (x - b) (x - c) \dots (x - z)\).
Prove that the polynomial \(P (x)\) is divisible by its derivative if and only if \(P (x)\) has the form \(P(x) = a_n(x - x_0)^n\).
A class contains 33 pupils, who have a combined age of 430 years. Prove that if we picked the 20 oldest pupils they would have a combined age of no less than 260 years. The age of any given pupil is a whole number.
Find the largest value of the expression \(a + b + c + d - ab - bc - cd - da\), if each of the numbers \(a\), \(b\), \(c\) and \(d\) belongs to the interval \([0, 1]\).
A polynomial of degree \(n > 1\) has \(n\) distinct roots \(x_1, x_2, \dots , x_n\). Its derivative has the roots \(y_1, y_2, \dots , y_{n-1}\). Prove the inequality \[\frac{x_1^2 + \dots + x_n^2}{n}> \frac{y_1^2 + \dots + y_n^2}{n}.\]
The number \(x\) is such a number that exactly one of the four numbers \(a = x - \sqrt{2}\), \(b = x-1/x\), \(c = x + 1/x\), \(d = x^2 + 2\sqrt{2}\) is not an integer. Find all such \(x\).