Harmonic Progressions

If in a $∆ABC, \sin A, \sin B, \sin C$ are in $\mathrm{A}.\mathrm{P}.,$ then –

H.M. between two numbers is $4$. The A.M. $A$ and the G.M. $G$ between them satisfy the relation $2A+G^2=27$. The numbers are

${\log }_{3}2$ , ${\log }_{6}2$ , ${\log }_{12}2$ are in

If for the harmonic progression, $t_7=\frac{1}{10}, t_{12}=\frac{1}{25}$, then $t_{20}=$

If $\cos \left(x-\frac{\pi }{3}\right), \cos x, \cos \left(x+\frac{\pi }{3}\right)$ are in a harmonic progression, then $\cos x=$

The number of values of $x$ such that the three terms $x,\left[x\right],\left\{x\right\}$ are in H.P., where $\left[·\right],\left\{·\right\}$ represents greatest integer function and fractional part function respectively, is/are

Let the harmonic mean and geometric mean of two positive number be in the ratio $4: 5$. then the ratio of two number is $\lambda (\lambda <1),$ find $11\lambda$. 

If the roots of the equation $10x^3-c{x}^2-54x-27=0$ are in harmonic progression, then the value of $c$ must be equal to

$H_1,H_2 \mathrm{a}\mathrm{r}\mathrm{e} 2 H.M{.}^{\prime }s$ between $a,b \mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n} \frac{H_1+H_2}{H_1·H_2}=$

If $2\left(y-a\right)$ is the harmonic mean between $y-x$ and $y-z$, then $x-a, y-a$ and $z-a$ are in

Let $\mathrm{ }a,b,c\mathrm{ }$ be non-zero real numbers such that $\mathrm{ }{a}^2,b^2,c^2\mathrm{ }$ are in harmonic mean and $a, b, c$ are in $A.P$, then

If non-zero numbers $\mathrm{ }\mathrm{a},\mathrm{b},\mathrm{c}\mathrm{ }$ are in H.P, then the straight line $\frac{\mathrm{x}}{\mathrm{a}}+\frac{\mathrm{y}}{\mathrm{b}}+\frac{1}{\mathrm{c}}=0$ always passes through a fixed point. That point is

Let $a_1, a_2, a_3, a_4 and a_5$ be such that $a_1, a_2 and a_3$ are in A.P., $a_2, a_3 and a_4$ are in G.P., and $a_3, a_4 and a_5$ are in H.P. Then $a_1, a_3 and a_5$ are in

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

Let ${\mathrm{I}}_{\mathrm{n}}=\mathrm{ }\underset{0}{\overset{\mathrm{\pi }/4}{\int }}{\tan }^{\mathrm{n}}\mathrm{x}\mathrm{ }\mathrm{d}\mathrm{x}$ . Then ${\mathrm{I}}_2+\mathrm{ }{\mathrm{I}}_4,\mathrm{ }{\mathrm{I}}_3+\mathrm{ }{\mathrm{I}}_5,\mathrm{ }{\mathrm{I}}_4+\mathrm{ }{\mathrm{I}}_6,\mathrm{ }{\mathrm{I}}_5+\mathrm{ }{\mathrm{I}}_7,\mathrm{ }\dots \dots$ are in -

If $\mathrm{x},\mathrm{ }\mathrm{y},\mathrm{ }\mathrm{z}$ are in $\mathrm{H}\mathrm{P}$ , then the value of expression ${\log }_{\mathrm{e}}\left(\mathrm{x}\mathrm{ }+\mathrm{ }\mathrm{z}\right)+{\log }_{\mathrm{e}}\left(\mathrm{x}\mathrm{ }-\mathrm{ }2\mathrm{y}\mathrm{ }+\mathrm{ }\mathrm{z}\right)$ will be -

If $a,b,c$ are in $\mathrm{H}.\mathrm{P}.,\mathrm{ }b,c,d$ are in $\mathrm{G}.\mathrm{P}.$ and $c,d,e$ are in $\mathrm{A}.\mathrm{P}.$ then value of $e$ is -

For a positive integer $\mathrm{n}$ let

$\mathrm{a}\left(\mathrm{n}\right)=\mathrm{ }1\mathrm{ }+\mathrm{ }\frac{1}{2}+\frac{1}{3}\mathrm{ }+\mathrm{ }\frac{1}{4}\dots \dots .+\frac{1}{2^{\mathrm{n}}-1}$ then -

Harmonic mean of the reciprocal of even numbers from $2 \text{to} 200$ is