If $a, b,$ and $c$ are in both Arithmetic Progression $(AP)$ and Geometric Progression $(GP)$,then........

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 $a^2(b + c), b^2(c + a), c^2(a + b)$ are in Arithmetic Progression $(AP)$,then $a, b, c$ are in which progression?

Given a sequence of $4$ numbers,the first three of which are in $G.P.$ and the last three are in $A.P.$ with a common difference of $6$. If the first and last terms in this sequence are equal,then the last term is:

If $a, b, c$ are in $A.P.$ and $a^2, b^2, c^2$ are in $H.P.$,then

The difference between two numbers is $48$ and the difference between their arithmetic mean and geometric mean is $18$. The larger of the two numbers is: