What is the minimum value of the sum of the real numbers $a^{-5}, a^{-4}, 3a^{-3}, 1, a^8$,and $a^{10}$ where $a > 0$?

Let $\frac{1}{16}, a$ and $b$ be in $G.P.$ and $\frac{1}{a}, \frac{1}{b}, 6$ be in $A.P.,$ where $a, b > 0$. Then $72(a + b)$ is equal to ...... .

Let $G$ be the geometric mean of two positive numbers $a$ and $b,$ and $M$ be the arithmetic mean of $\frac{1}{a}$ and $\frac{1}{b}.$ If $\frac{1}{M} : G$ is $4:5,$ then $a:b$ can be

If the arithmetic and geometric means of $a$ and $b$ are $A$ and $G$ respectively,then the value of $A - G$ is

If $2(y - a)$ is the $H.M.$ between $y - x$ and $y - z$,then $x - a, y - a, z - a$ are in

In a $G.P.$,the sum of three numbers is $14$. If $1$ is added to the first two numbers and subtracted from the third number,the series becomes an $A.P.$. Then,the greatest number is: