Harmonic Progression

If $\mathrm{A}.\mathrm{M}.$ and $\mathrm{G}.\mathrm{M}.$ of two numbers is $10$ and $8$ respectively then find their $\mathrm{H}.\mathrm{M}. :$

If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{AP} \mathrm{b}, \mathrm{c}, \mathrm{d}$ are in $\mathrm{G}.\mathrm{P}.$ and $\mathrm{c}, \mathrm{d}, \mathrm{e}$ are in $\mathrm{H}.\mathrm{P}.,$ then which one of the following is true :

Which one of the following is true about $\mathrm{A}.\mathrm{M}., \mathrm{G}.\mathrm{M}.$ and $\mathrm{H}.\mathrm{M}.$

For a positive integer $\mathrm{n},$ let $\mathrm{a}\left(\mathrm{n}\right)=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\dots +\frac{1}{\left(2^{\mathrm{n}}\right)-1}.$ Then, which of the following are correct?

(i) $\mathrm{a}\left(100\right)\le 100$

(ii) $\mathrm{a}\left(100\right)>100$
(iii) $\mathrm{a}\left(200\right)\le 100$
(iv) $\mathrm{a}\left(200\right)>100$

Let ${\mathrm{a}}_1,{\mathrm{a}}_2,\dots ,{\mathrm{a}}_{10}$ be in A.P. and ${\mathrm{h}}_1,{\mathrm{h}}_2,\dots ,{\mathrm{h}}_{10}$ be in H.P. If ${\mathrm{a}}_1={\mathrm{h}}_1=2$ and ${\mathrm{a}}_{10}={\mathrm{h}}_{10}=3,$ then ${\mathrm{a}}_4{\mathrm{h}}_7$ is

The harmonic mean of the roots of the equation $\left(5+\sqrt{2}\right)x^2-\left(4+\sqrt{5}\right)x+\left(8+2\sqrt{5}=0\right)$ is