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

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

Let $a,b,c$ and $d$ be positive real numbers such that $a+b+c+d=11$. If the maximum value of $a^5{b}^3{c}^{2}d$ is $3750\beta$, then the value of $\beta$ is

Minimum value of $\left(a+b+c+d\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\right)$ is 

If $x,y,z>0$ and $x+y+z=1$ then $\frac{xyz}{\left(1-x\right)\left(1-y\right)\left(1-z\right)}$ is necessarily.

If $xyz=\left(1-x\right)\left(1-y\right)\left(1-z\right)$ where, $0\le x,y,z\le 1$, then the minimum value of $x\left(1-z\right)+y\left(1-x\right)+z\left(1-y\right)$ is:

The minimum value of $\frac{\left(x^2+x+1\right)\left(y^2+y+1\right)}{xy}$ where $x,y >0$ is

For a distinct complex numbers $z_1,z_2,z_3\dots ,z_n\left(n\ge 3\right)$, the min value of $\frac{{\left|z_2-z_1\right|}^2+{\left|z_3-z_2\right|}^2+\cdots +{\left|z_n-z_{n-1}\right|}^2}{{\left|z_n-z_1\right|}^2}$ 

If $x,y,z$ are positive integers, then $\left(x+y\right)\left(y+z\right)\left(z+x\right)$ is

If $a,b,c$ are non-zero real numbers, then the minimum value of the expression $\frac{\left(a^8+4a^4+1\right)\left(b^8+3b^4+1\right)\left(c^2+2c+2\right)}{a^4{b}^4}$ equals

If $x+y=15$, and $x^2{y}^3$ is maximum, then

The sum of square of any real positive quantities and its reciprocal is never less

If $\overrightarrow{e}=l\hat{i}+m\hat{j}+n\hat{k}$ is a unit vector and $l=\frac{1}{3},$ then maximum value of $lmn$ is

Find the number of real ordered pair(s) $\left(x,y\right)$ for which: ${16}^{x^2+y}+{16}^{x+y^2}=1$

If A.M of two numbers is $11$ and their G.M. is $2\sqrt{30}$, find the two numbers.
 

If A.M of two numbers be twice their G.M then the numbers are in the ratio.

For the two data points $4, 5$, find the relationship between AM, GM, HM, prove that, AM > GM > HM.

For the two data points $10, 12$, find the relationship between AM, GM, HM, prove that, AM > GM > HM.

Find the value of geometric mean GM, if the arithmetic mean AM is $7$, and harmonic mean HM is $\frac{48}{7}$.