If $a, b, c$ are three unequal numbers such that $a, b, c$ are in $A.P.$ and $b - a, c - b, a$ are in $G.P.$,then $a : b : c$ is

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

If $a, b, c$ are in $A.P.$ and $a, c - b, b - a$ are in $G.P.$ $(a \ne b \ne c)$,then $a:b:c$ is

Let $b_i > 1$ for $i = 1, 2, \ldots, 101$. Suppose $\log _e b_1, \log _e b_2, \ldots, \log _e b_{101}$ are in Arithmetic Progression $(A.P.)$ with the common difference $\log _e 2$. Suppose $a_1, a_2, \ldots, a_{101}$ are in $A.P.$ such that $a_1 = b_1$ and $a_{51} = b_{51}$. If $t = b_1 + b_2 + \cdots + b_{51}$ and $s = a_1 + a_2 + \cdots + a_{51}$,then:

If the $A.M.$ of two numbers is greater than the $G.M.$ of the numbers by $2$ and the ratio of the numbers is $4:1$,then the numbers are

If the ratio of two numbers is $9:1$,then the ratio of their geometric mean to their harmonic mean is