A.M., G.M., H.M. and their Relations

If $a_1,a_2,.....a_n$ are positive real numbers whose product is a fixed number $e$, the minimum value of $a_1+a_2+a_3+.......+a_{n-1}+2a_n$ is

If $a_1, a_2, .......a_n$ are positive real numbers whose product is a fixed number $e$, the minimum value of $a_1+a_2+ .......+2a_n$ is

Let $\sin \beta$ be the $GM$ of $\sin \alpha$ and $\cos \alpha ; \tan \gamma$ be the $AM$ of $\sin \alpha$ and $\cos \alpha$.

What is the value of $\sec 2\gamma$?

Let $\sin \beta$ be the $GM$ of $\sin \alpha$ and $\cos \alpha ; \tan \gamma$ be the $AM$ of $\sin \alpha$ and $\cos \alpha$.

What is $\cos 2\beta$ equal to?

A quadratic equation is given by

$\left(3+2\sqrt{2}\right)x^2-\left(4+2\sqrt{3}\right)x+\left(8+4\sqrt{3}\right)=0$

What is the GM of the roots of the equation?

A quadratic equation is given by

$\left(3+2\sqrt{2}\right)x^2-\left(4+2\sqrt{3}\right)x+\left(8+4\sqrt{3}\right)=0$

What is the HM of the roots of the equation?

If $G$ is the geometric mean of numbers $1, 2, 2^2, 2^3,\dots , 2^{n-1}$, then what is the value of $1+2{\log }_{2}G$?

Given that $m\left(\theta \right)={\cot }^2\theta +n^2 {\tan }^2\theta +2n$, where $n$ is a fixed positive real number.

Under what condition does $m$ attain the least value?

Given that $m\left(\theta \right)={\cot }^2\theta +n^2 {\tan }^2\theta +2n$, where $n$ is a fixed positive real number.

What is the least value of $m\left(\theta \right)$?

Let $x$ and $y$ be positive real numbers. What is the smallest possible value of $\frac{16}{x}+\frac{108}{y}+xy$?

For three positive numbers $a, b \& c$, the minimum value of $\frac{\left(a+b+c\right)\left(a^2+b^2+c^2\right)}{abc}$ is equal to

If $a,b,c$ are positive real numbers, then minimum value of $\frac{\left(a+b+c\right)\left(a^2+b^2+c^2\right)}{abc}$ is

Let $x$ and $y$ be two positive real numbers such that $x+y=1$. Then the minimum value of $\frac{1}{x}+\frac{1}{y}$ is-

The number of different possible values for the sum $x+y+z,$ where $x, y, z$ are real number such that $x^4+4y^4+16z^4+64=32xyz$ is

The number of three digit numbers $\overline{abc}$ such that the arithmetic mean of $b \& c$ and the square of their geometric mean are equal is

The minimum value of sum of real numbers $a^{-6}, 2a^{-4}, 2a^{-3},1$ and $2a^{10}$ with $a>0$ is equal to

In a triangle $ABC$, if $\tan A+\tan B+\tan C=8,$ then the least value of ${\tan }^{2}A+{\tan }^{2}B+{\tan }^{2}C$ is

The minimum value of the expression ${\left(t^2+1-\alpha \right)}^2+(2t-\alpha -4{)}^2, (t, \alpha \in R)$ is

If ${\alpha }^2+{\beta }^2+{\gamma }^2=1,$ then highest integral value of $\alpha \beta +\beta \gamma +\alpha \gamma$ is

If $a,b,c\in R^{+}$ such that $a+b+c=27$ and maximum value of $a^2{b}^3{c}^4=2^m\times 3^n,$ then the sum of six A.M.'s between $m$ and $n$ is