If $A$,$G$,and $H$ are the arithmetic,geometric,and harmonic means between two positive real numbers,respectively,then:

If $a_1, a_2, \dots, a_n$ are positive real numbers such that $a_1 \cdot a_2 \cdot \dots \cdot a_n = 1$,then their sum is:

The sum of three decreasing numbers in $A.P.$ is $27$. If $-1, -1, 3$ are added to them respectively,the resulting series is in $G.P.$ The numbers are

If $x = \sum_{n=0}^{\infty} a^{n}$,$y = \sum_{n=0}^{\infty} b^{n}$,$z = \sum_{n=0}^{\infty} c^{n}$,where $a, b, c$ are in $A.P.$ and $|a| < 1, |b| < 1, |c| < 1$,$abc \neq 0$,then:

If the Arithmetic Mean $(AM)$ $= 16$ and the Harmonic Mean $(HM)$ $= \frac{63}{4}$ for any two numbers,what will be the Geometric Mean $(GM)$?

The common difference of an $A.P.$ whose first term is unity and whose second,tenth and thirty-fourth terms are in $G.P.$ is