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.
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 \((x - a) (x - b) (x - c) \dots (x - z)\).
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.
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 \(x\).
The numbers \(x\), \(y\) and \(z\) are such that all three numbers \(x + yz\), \(y + zx\) and \(z + xy\) are rational, and \(x^2 + y^2 = 1\). Prove that the number \(xyz^2\) is also rational.
The numerical function \(f\) is such that for any \(x\) and \(y\) the equality \(f (x + y) = f (x) + f (y) + 80xy\) holds. Find \(f(1)\) if \(f(0.25) = 2\).
Valentina added a number (not equal to 0) taken to the power of four and the same number to the power two and reported the result to Peter. Can Peter determine the unique number that Valentina chose?
Is it possible to:
a) load two coins so that the probability of “heads” and “tails” were different, and the probability of getting any of the combinations “tails, tails,” “heads, tails”, “heads, heads” be the same?
b) load two dice so that the probability of getting any amount from 2 to 12 would be the same?