Problems

How many rational terms are contained in the expansion of

a) $(\sqrt{2}+\sqrt[4]{3}{)}^{100}$;

b) $(\sqrt{2}+\sqrt[3]{3}{)}^{300}$?

Which term in the expansion $(1+\sqrt{3}{)}^{100}$ will be the largest by the Newton binomial formula?

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 $r\ge 2$.

Suppose that there are 15 prime numbers forming an arithmetic progression with a difference of $d$. Prove that $d>30,000$.

Could it be that a) $\sigma (n)>3n$; b) $\sigma (n)>100n$?

Prove that in any infinite decimal fraction you can rearrange the numbers so that the resulting fraction becomes a rational number.

Prove that there are infinitely many composite numbers among the numbers $\lfloor 2^k\sqrt{2}\rfloor$ ($k=0,1,\dots$).

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.