If $a, b, c$ are in $G.P.$ and $a + x, b + x, c + x$ are in $H.P.$,then the value of $x$ is ($a, b, c$ are distinct numbers).

Let $A_1, G_1, H_1$ be the arithmetic,geometric,and harmonic means of two distinct positive numbers. For $n \geq 2$,let $A_n, G_n, H_n$ be the arithmetic,geometric,and harmonic means of $A_{n-1}$ and $H_{n-1}$ respectively. Which of the following statements is true?

Consider two positive numbers $a$ and $b$. If the arithmetic mean of $a$ and $b$ exceeds their geometric mean by $\frac{3}{2}$ and the geometric mean of $a$ and $b$ exceeds their harmonic mean by $\frac{6}{5}$,then the absolute value of $(a^2 - b^2)$ is equal to

If $A.M.$ and $G.M.$ of two positive numbers $a$ and $b$ are $10$ and $8$ respectively,find the numbers.

Let $x_{1}, x_{2}$ be the roots of $x^{2}-3x+a=0$ and $x_{3}, x_{4}$ be the roots of $x^{2}-12x+b=0$. If $x_{1} < x_{2} < x_{3} < x_{4}$ and $x_{1}, x_{2}, x_{3}, x_{4}$ are in $GP$,then $ab$ equals:

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 $a > 0$?