Let ${a_1},{a_2},{a_3}$ be any positive real numbers, then which of the following statement is not true
$3{a_1}{a_2}{a_3} \le a_1^3 + a_2^3 + a_3^3$
$\frac{{{a_1}}}{{{a_2}}} + \frac{{{a_2}}}{{{a_3}}} + \frac{{{a_3}}}{{{a_1}}} \ge 3$
$({a_1} + {a_2} + {a_3})\,\left( {\frac{1}{{{a_1}}} + \frac{1}{{{a_2}}} + \frac{1}{{{a_3}}}} \right) \ge 9$
$({a_1} + {a_2} + {a_3})\,{\left( {\frac{1}{{{a_1}}} + \frac{1}{{{a_2}}} + \frac{1}{{{a_3}}}} \right)^3} \le 27$
If $\frac{{a + bx}}{{a - bx}} = \frac{{b + cx}}{{b - cx}} = \frac{{c + dx}}{{c - dx}}(x \ne 0)$, then $a,\;b,\;c,\;d$ are in
If $a$ and $b$ are two different positive real numbers, then which of the following relations is true
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
Let $a, b, c > 1, a^3, b^3$ and $c^3$ be in $A.P.$, and $\log _a b$, $\log _c a$ and $\log _b c$ be in G.P. If the sum of first $20$ terms of an $A.P.$, whose first term is $\frac{a+4 b+c}{3}$ and the common difference is $\frac{a-8 b+c}{10}$ is $-444$, then abc is equal to
If $A$ is the $A.M.$ of the roots of the equation ${x^2} - 2ax + b = 0$ and $G$ is the $G.M.$ of the roots of the equation ${x^2} - 2bx + {a^2} = 0,$ then