How many nine-digit numbers exist, the sum of the digits of which is even?
Calculate the following sums:
a) \(\binom{5}{0} + 2\binom{5}{1} + 2^2\binom{5}{2} + \dots +2^5\binom{5}{5}\);
b) \(\binom{n}{0} - \binom{n}{1} + \dots + (-1)^n\binom{n}{n}\);
c) \(\binom{n}{0} + \binom{n}{1} + \dots + \binom{n}{n}\).
In the expansion of \((x + y)^n\), using the Newton binomial formula, the second term was 240, the third – 720, and the fourth – 1080. Find \(x\), \(y\) and \(n\).
Here is a fragment of the table, which is called the Leibniz triangle. Its properties are “analogous in the sense of the opposite” to the properties of Pascal’s triangle. The numbers on the boundary of the triangle are the inverses of consecutive natural numbers. Each number is equal to the sum of two numbers below it. Find the formula that connects the numbers from Pascal’s and Leibniz triangles.
Find the sums of the following series:
a) \({\frac {1} {1 \times 2}} + {\frac {1} {2 \times 3}} + {\frac {1} {3 \times 4}} + {\frac {1} {4 \times 5}} + \dots\);
b) \({\frac {1} {1 \times 2 \times 3}} + {\frac {1} {2 \times 3 \times 4}} + {\frac {1} {3 \times 4 \times 5}} + {\frac {1} {4 \times 5 \times 6}} + \dots\);
c) \({\frac {0!} {r!}} + {\frac {1!} {(r-1)!}} + {\frac {2!} {(r-2) !}} + {\frac {3!} {(r-3)!}} + \dots\) for \(r \geq 2\).
Could it be that a) \(\sigma(n) > 3n\); b) \(\sigma(n) > 100n\)?
Prove that for a real positive \(\alpha\) and a positive integer \(d\), \(\lfloor \alpha / d\rfloor = \lfloor \lfloor \alpha\rfloor / d\rfloor\) is always satisfied.
Prove the irrationality of the following numbers:
a) \(\sqrt{3}{17}\)
b) \(\sqrt{2} + \sqrt{3}\)
c) \(\sqrt{2} + \sqrt{3} + \sqrt{5}\)
d) \(\sqrt{3}{3} - \sqrt{2}\)
e) \(\cos 10^{\circ}\)
f) \(\tan 10^{\circ}\)
g) \(\sin 1^{\circ}\)
h) \(\log_{2}3\)
Is it possible for
a) the sum of two rational numbers irrational?
b) the sum of two irrational numbers rational?
c) an irrational number with an irrational degree to be rational?
One of the roots of the equation \(x^2 + ax + b = 0\) is \(1 + \sqrt 3\). Find \(a\) and \(b\) if you know that they are rational.