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 $\frac{{\left(\sqrt{a}\right)}^{\left(n+2\right)}+{\left(\sqrt{b}\right)}^{\left(n+2\right)}}{{\left(\sqrt{a}\right)}^n+{\left(\sqrt{b}\right)}^n}$ 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

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

Two numbers $x$ and $y$ have arithmetic mean $9$ and geometric mean $4.$ Then, $x$ and $y$ are the roots of

Marks of a student in Maths and science is $147$ and $192$ out of $200$. Find the geometric mean of the marks obtained by student.

What is the geometric mean of $4$ and $16$?

Find the geometric mean of $4$ and $3$.

If $\frac{a^{n-\frac{1}{2}}+b^{n-\frac{1}{2}}}{a^{n+\frac{1}{2}}+b^{n+\frac{1}{2}}}$ is geometric mean of $\frac{1}{a}, \frac{1}{b}\left(a\ne b\right),$ then the quadratic equation $b{x}^2+nx+b=0$ has

If $n$ geometric means be inserted between $a$ and $b$ then their product is 

If ${\log }_{10}a, {\log }_{10}b, {\log }_{10}c$ are in $A.P.$, then $a,b,c$ must be in:

Consider the series $21, 22, 23,..., k-1, k$ where $k$ is a $3$ digit number. Given that the $A.M.$ and $G.M.$ of first and last term of the given series is contained in the series itself, then possible number of values of $k$ are equal to:

If geometric mean between $x$ and $y$ is $G$, then find the value of $\frac{1}{{\mathrm{G}}^2-{\mathrm{x}}^2}+\frac{1}{{\mathrm{G}}^2-{\mathrm{y}}^2}$.

Cotyledons are also called-

Consider the series $21, 22, 23,...,k-1,k$ where $k$ is a three-digit number. Given that the $A.M$ and $G.M$of first and last term of the given series is contained in the series itself, then possible number of values of $k$ are

If the AM between $\mathrm{‘}a\mathrm{’}\mathrm{ }\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\mathrm{‘}b\mathrm{’}\mathrm{ }$ is twice as greater as the GM, then the value of $\frac{a}{b}$ is

$\sin A,\sin B,\sin C\mathrm{ }$ are in G.P, $\mathrm{ }$ roots of $\mathrm{ }{x}^2+2xcotB+1=0$ are always:

If $m \text{is the A. M.}$ of two distinct real numbers $l \text{and}\mathrm{ }n\left(\mathrm{ }l,\mathrm{ }n>1\right)$, and $\mathrm{ }{G}_1\mathrm{ },G_2 \text{and} G_3\mathrm{ }$ are three geometric means between $1$ and $n,$ then value of $\mathrm{ }\mathrm{ }{G}_1^4+G_3^4+2G_2^4$ is

If $A_1,A_2$ are two A.M.s $G_1,G_2$ are two G.M.s and $H_1,H_2$ are two H.M.s between two numbers, then $\frac{A_1+A_2}{H_1+H_2}$ is equal to