If the Arithmetic Mean of $a$ and $b$ is $\frac{a^{n+1} + b^{n+1}}{a^n + b^n}$,then $n = \dots$

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

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

If $a, b, c$ are in $A.P.$ and $a, b, d$ are in $G.P.$,then $a, a - b, d - c$ will be in

Three positive numbers form an increasing $G.P.$ If the middle term in this $G.P.$ is doubled,the new numbers are in $A.P.$ Then the common ratio of the $G.P.$ is:

If $a, b, c$ are in geometric progression,then in which progression are $\frac{d}{a}, \frac{e}{b}, \frac{f}{c}$ such that the equations $ax^2 + 2bx + c = 0$ and $dx^2 + 2ex + f = 0$ have common roots?