The $AM$ of two given positive numbers is $2$. If the larger number is increased by $1$, the $GM$ of the numbers becomes equal to the $AM$ of the given numbers. Then the $HM$ of the given numbers is
 

In a GP, the product of the first four terms is 4 and the second term is the reciprocal of the fourth term. The sum of the GP upto infinite terms is:

(i) $8$

(ii) $-8$

(iii) $\frac{8}{3}$

(iv) $-\frac{8}{3}$

Let ${\mathrm{a}}_{\mathrm{n}}=$ product of the first $\mathrm{n}$  natural numbers. Then, for all $n\in \mathrm{N}$,

(i) ${\mathrm{n}}^{\mathrm{n}}\ge {\mathrm{a}}_{\mathrm{n}}$

(ii) ${\left(\frac{\mathrm{n}+1}{2}\right)}^{\mathrm{n}}\ge \mathrm{n}!$

(iii) ${\mathrm{n}}^{\mathrm{n}}\ge {\mathrm{a}}_{\mathrm{n}}$

 (iv) None of these