If $a,\;b,\;c$ are in $G.P.$ and $\log a - \log 2b,\;\log 2b - \log 3c$ and $\log 3c - \log a$ are in $A.P.$, then $a,\;b,\;c$ are the length of the sides of a triangle which is

The sum of three consecutive terms in a geometric progression is $14$. If $1$ is added to the first and the second terms and $1$ is subtracted from the third, the resulting new terms are in arithmetic progression. Then the lowest of the original term is

If the first and ${(2n - 1)^{th}}$ terms of an $A.P., G.P.$ and $H.P.$ are equal and their ${n^{th}}$ terms are respectively $a,\;b$ and $c$, then

Three numbers form a $G.P.$ If the ${3^{rd}}$ term is decreased by $64$, then the three numbers thus obtained will constitute an $A.P.$ If the second term of this $A.P.$ is decreased by $8$, a $G.P.$ will be formed again, then the numbers will be

Let $m$ be the minimum possible value of $\log _3\left(3^{y_1}+3^{y_2}+3^{y_3}\right)$, where $y _1, y _2, y _3$ are real numbers for which $y _1+ y _2+ y _3=9$. Let $M$ be the maximum possible value of $\left(\log _3 x _1+\log _3 x _2+\log _3 x _3\right)$, where $x_1, x_2, x_3$ are positive real numbers for which $x_1+x_2+x_3=9$. Then the value of $\log _2\left(m^3\right)+\log _3\left(M^2\right)$ is. . . . . . 

Let $x, y, z$  be positive real numbers such that $x + y + z = 12$ and  $x^3y^4z^5 = (0. 1 ) (600)^3$. Then $x^3 + y^3 + z^3$ is equal to