Harmonic Progression

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

If $H$ is the harmonic mean of numbers $1, 2, 2^2, 2^3,\dots , 2^{n-1}$, then what is $n/H$ equal to?

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 $\cos x=$

$H_1,H_2 \mathrm{a}\mathrm{r}\mathrm{e} 2 H.M{.}^{\prime }s$ 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

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

The largest positive term of $\mathrm{H}.\mathrm{P}$ . whose first two terms are $\frac{2}{5}$ and $\frac{12}{23}$ is

If ${\mathrm{H}}_1,\mathrm{ }{\mathrm{H}}_2,\mathrm{ }{\mathrm{H}}_3\mathrm{ }\dots \dots ..{\mathrm{H}}_{2\mathrm{n}+1}$ are in $\mathrm{H}.\mathrm{P}$ . then

$\sum _{\mathrm{i}=1}^{2\mathrm{n}}{\left(-1\right)}^{\mathrm{i}}\mathrm{ }\left(\frac{{\mathrm{H}}_{\mathrm{i}}+{\mathrm{H}}_{\mathrm{i}+1}}{{\mathrm{H}}_{\mathrm{i}}-{\mathrm{H}}_{\mathrm{i}+1}}\right)$ is equal to -

If $2(y–a)$ is the $H.M.$ between $y – x$ and $y – z$ , then $x-a,\mathrm{ }\mathrm{ }y-a,\mathrm{ }\mathrm{ }z-a$ are in

If a, b, c are in H.P., then the straight line $\frac{x}{a}+\frac{y}{b}+\frac{1}{c}=0$ always passes through a fixed point and that point is

$H_1$, $H_2$ are $2$ $H.M.\text{'}s$ between $a$, $b$ then $\frac{H_1+H_2}{H_1\cdot H_2}=$