The sum of three decreasing numbers in $A.P.$ is $27$. If $ - 1,\, - 1,\,3$ are added to them respectively, the resulting series is in $G.P.$ The numbers are

Two sequences $\{ {t_n}\} $ and $\{ {s_n}\} $ are defined by ${t_n} = \log \left( {\frac{{{5^{n + 1}}}}{{{3^{n - 1}}}}} \right)\,,\,\,{s_n} = {\left[ {\log \left( {\frac{5}{3}} \right)} \right]^n}$, then

The product of three consecutive terms of a $G.P.$ is $512$. If $4$ is added to each of the first and the second of these terms, the three terms now form an $A.P.$ Then the sum of the original three terms of the given $G.P.$ is

If $G.M. = 18$ and $A.M. = 27$, then $H.M.$ is

Let $x, y>0$. If $x^{3} y^{2}=2^{15}$, then the least value of $3 x +2 y$ 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 $s=a_1+a_2+\cdots+t_{65}$, then