Relationship Between Arithmetic Mean and Geometric Mean

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

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

If $x>m\; ,\; y>n\; ,\; z>\text{r}(x,\; y,\; z\; >\; 0)$ such that $\left|\begin{array}{ccc}\text{x}& \text{n}& \text{r}\\ \text{m}& \text{y}& \text{r}\\ \text{m}& \text{n}& \text{z}\end{array}\right|=0$.

 The greatest value of $\frac{\text{xyz}}{\left(\text{x}-\text{m}\right)\left(\text{y}-\text{n}\right)\left(\text{z}-\text{r}\right)}$  is 

If $x>m\; ,\; y>n\; ,\; z>\text{r}(x,\; y,\; z\; >\; 0)$ such that $\left|\begin{array}{ccc}\text{x}& \text{n}& \text{r}\\ \text{m}& \text{y}& \text{r}\\ \text{m}& \text{n}& \text{z}\end{array}\right|=0$.

The value of $\frac{\text{m}}{\text{x}-\text{m}}+\frac{\text{y}}{\text{y}-\text{n}}+\frac{\text{r}}{\text{z}-\text{r}}$  is

If $x>\mathrm{m},\mathrm{y}>\mathrm{n},\mathrm{z}>r(\mathrm{x},\mathrm{y,}\mathrm{z\; >\; 0)}$ such that $\left|\begin{array}{ccc}x& n& r\\ m& y& r\\ m& n& z\end{array}\right|=0$. The value of $\frac{x}{x-m}+\frac{y}{y-n}+\frac{z}{z-r}$, is

If $x, y\in (0,\infty )$, then the least possible value of $x^{100}+\frac{100}{x}+y^{50}+\frac{50}{y}$ is

If the first and ${\left(2n-1\right)}^{th}$terms of an $A.P, G.P \text{and} H.P$ with positive terms are equal and their $n^{th}$ terms are $a, b, c$ respectively, then which of the following options must be correct: 

Let $a, x, b$ be in $AP ; a, y, b$ are in $GP$ and $a, z, b$ are in $HP$ where $a$ and $b$ are distinct positive real number. If $x=y+2$ and $a=5z$, then :

Let $1, abc,$ $a^2{b}^2{c}^2$ are in $A.P.$ $(a, b, c>0),$ then minimum value of $27 a+8 b+125 c$ is :

Statement $1$: $27abc\ge {\left(a+b+c\right)}^3$ and $3a+4b+5c=12$ then $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}=10,$ where$a, b, c$  positive real number.
Statement $2$: For positive numbers, $\mathrm{AM}>\mathrm{GM}$.

If $a, b \& c$ are positive real numbers and $\{\left(1+a\right)\left(1+b\right)\left(1+c\right){\}}^7>k^7{a}^4{b}^4{c}^4$, then maximum integer value of $k$ is

The product of $n$ positive numbers is unity. Then, their sum is

If $x, y$ is positive real numbers then maximum value of $\frac{x^m{y}^n}{\left(1 + x^{2m}\right)\left(1 + y^{2n}\right)}$ is

If positive number $x,y,z$ are in $A.P.,$ then the minimum value of $\frac{x+y}{2y-x}+\frac{y+z}{2y-z}$ is equal to

If where $a_i\ge 0,i=1,2,3,....n$ and $\sum _{i=1}^n{a}_i=20$ and if the greatest possible value of $\sum _{i=1}^{n-1}{a}_i\left(a_{i+1}\right)$ is S then S is equal to

If $f\left(\theta \right)=2({\sec }^2\theta +{\cos }^2\theta )$$,$ then its value always

If $x$ and $y$ are positive real numbers then minimum value of $x^{100}+\frac{100}{x}+y^{50}+\frac{50}{y}$ is

If the three positive real numbers $a, b, c$ are in A.P. and $abc = 64$, then the minimum value of $b$ is