Powers

Here is a problem with indices:

1) Show that $2^2 3^3>2^3 3^2$.

* 2) Show that: $$\left(2\times3\times4\times5\times\cdots\times100\times101\right)^2<2\times3^3\times4^3\times5^5\times6^5\times\cdots\times100^{99}\times101^{101}$$
** 3) Prove that:
$$\left(2\times3\times4\times5\times\cdots\times100\times101\right)^2 > 2^3\times3\times4\times5^3\times6^3\times7\times\cdots\times100\times101^3$$

Continued Fractions

Many mathematicians know that: $$\phi = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \ddots}}}}$$ as a continued fraction. But do you know why?

Here are some problems on the theme 'continued':

1) Evaluate $\sqrt{3\sqrt{3\sqrt{3\cdots}}}$

2) Evaluate $\sqrt{3+\sqrt{3+\sqrt{3+\cdots}}}$

3) Evaluate $\frac{\frac{\frac{a+1}{2}+1}{2}+1}{2}+1\cdots$