Geometric Mean

Let $A_1$ and $A_2$ be two arithmetic means and $G_1, G_2$ and $G_3$ be three geometric means of two distinct positive numbers. Then $G_1^4+G_2^4+G_3^4+G_1^2{G}_3^2$ is equal to

If $G$ be the geometric mean between two given numbers and $A_1, A_2$ be the arithmetic means between them. Then, $G^2$

The product of three geometric means between $4$ and $\frac{1}{4}$ is$?$

Find the geometric mean of $133, 141, 125, 173, 182$

If $\frac{{\left(\sqrt{a}\right)}^{\left(n+2\right)}+{\left(\sqrt{b}\right)}^{\left(n+2\right)}}{{\left(\sqrt{a}\right)}^n+{\left(\sqrt{b}\right)}^n}$ is geometric mean between $a$ and $b\left(a,b>0\right)$ and $a\ne b$, then $n$ is

The arithmetic mean of two numbers is $30$ and geometric mean is $24$ find the two number

Insert four geometric means between the numbers $3 \text{and }96$?

Four geometric means between $4$and $972$are _____.

Let $a,b$ be distinct positive real numbers, whose geometric mean equals $\frac{a^{t-99}+b^{t-99}}{a^{t-100}+b^{t-100}}$. Then $t$ must equal

If $A$ and $G$ are respectively the A.M. and G.M. of two distinct positive numbers $a$ and $b$, find the value of $\frac{G^2-a^2}{aA-G^2}$.

If $\mathrm{G}$ is the G.M. between two positive numbers $a$ and $b$, show that

$\frac{1}{G^2-a^2}+\frac{1}{G^2-b^2}=\frac{1}{G^2}$.

If one G.M. G and two A.M.’s $A_1, A_2$ are inserted between two positive numbers, then show that $G2=(2A_2-A_1)(2A_1-A_2)$.

Between two positive numbers. one A.M. A also two G.M.'s $G_1,G_2$ are inserted. Prove that $\frac{G_1^2}{G_2}+\frac{G_2^2}{G_1}=2A$

One of two positive real numbers exceeds the other by $6$. A.M. between these two numbers exceeds their positive G.M. by unity. Find the numbers.

The product of two positive real numbers in $128$ and product of $n$ number of positive G.M.s between those two numbers is $(3n+2)\mathrm{th}$ power of $2$. Find the value of $n$.

If $x$ is the A.M. between $y$ and $z$, and $\frac{1}{x}$ is the A.M. of $\frac{1}{y}$and $\frac{1}{z}$, then show that $y,x,z$ are in G.P.

If $a,b,c$ are in A.P., $b,c,d$ are in G.P. and. $c,d,e$ are in A.P., prove that $a,c,e$ are in G.P.

If $a,b,c$ are in G.P. and $a^{{}_x^1}=b^{{}_y^1}=c^{{}_z^1}$, then show that $x,y,z$ are in A.P.

If $a$ is the A.M. of two positive numbers $b$ and $c$ and two G.M.'s between $b$ and $c$ are $G_1$ and $G_2$, prove that $G_1^3+G_2^3=2abc$.

If $a,b,c$ are in G.P. and the equation

$a{x}^2+2bx+c=0$ and $d{x}^2+2ex+f=0$ have a common root, then show that $\frac{d}{a},\frac{e}{b},\frac{f}{c}$ are in A.P.