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

The first term of an arithmetic progression is $1$. If the second,tenth,and thirty-fourth terms form a geometric progression,then the common difference of the arithmetic progression is:

If $a, b, c$ are in $A.P.$ and $b, c, d$ are in $H.P.$,then

If $a, b, c$ are in $A.P.$,then $2^{ax + 1}, 2^{bx + 1}, 2^{cx + 1}$ for $x \ne 0$ are in

If $a, b, c$ are in $G.P.$ and $a^{\frac{1}{x}} = b^{\frac{1}{y}} = c^{\frac{1}{z}} = k,$ prove that $x, y, z$ are in $A.P.$

If $a, b, c \in \mathbb{R}^+$ are such that $2a, b, 4c$ are in $A.P.$ and $c, a, b$ are in $G.P.$,then: