The harmonic mean of two numbers is $4$ and the arithmetic and geometric means satisfy the relation $2A + {G^2} = 27$, the numbers are
$6,\,3$
$5, \,4$
$5,\; - 2.5$
$ - 3,\;1$
Suppose $\log _a b+\log _b a=c$. The smallest possible integer value of $c$ for all $a, b>1$ is
If the ratio of $H.M.$ and $G.M.$ between two numbers $a$ and $b$ is $4:5$, then the ratio of the two numbers will be
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
If $a, b, c$ are in $A.P.;$ $b, c, d$ are in $G.P.$ and $\frac{1}{c}, \frac{1}{d}, \frac{1}{e}$ are in $A.P.$ prove that $a, c, e$ are in $G.P.$
If the ${p^{th}},\;{q^{th}}$ and ${r^{th}}$ term of a $G.P.$ and $H.P.$ are $a,\;b,\;c$, then $a(b - c)\log a + b(c - a)$ $\log b + c(a - b)\log c = $