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 $w,x,y,z$ are in $\mathrm{H}.\mathrm{P}.,$ then the value of $\frac{w^{-2}-z^{-2}}{2x^{-2}-2y^{-2}}$ is 

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 $\cos x=$

If $a, b, c$ are in A.P., $b,c,d$ are in G.P., and $c, d, e$ are in H.P. then $a, c, e$ are in

A line is drawn from $A(-4, 0)$ to intersect the curve $\frac{x^2}{8}+\frac{y^2}{4}=1$ at $P$ and $Q$ above $x$-axis. If $\frac{1}{AP}+\frac{1}{AQ}\ge \frac{\sqrt{3}}{2},$ then the maximum value of the slope of line is

An aeroplane flies around a square, the sides of which measure $100$ miles each. The aeroplane covers at a speed of $100 \mathrm{m} {\mathrm{h}}^{-1}$ the first side, at $200 \mathrm{m} {\mathrm{h}}^{-1}$ the second side, at $300 \mathrm{m} {\mathrm{h}}^{-1}$ the third side and $400 \mathrm{m} {\mathrm{h}}^{-1}$ the fourth side. The average speed of aeroplane around the square is (in $\mathrm{m} {\mathrm{h}}^{-1}$)

Let $a, b, c$ are positive real numbers such that  $p=a^{2}b+a{b}^2-a^{2}c-a{c}^2, q=b^{2}c+b{c}^2-a^{2}b-a{b}^2 \& r=a{c}^2+a^{2}c-c{b}^2-b{c}^2$ and the quadratic equation $p{x}^2+qx+r=0$ has equal roots, then $a, b \& c$ are in

If $a,b,c$ are in A.P.,  and $p, q, r$ are in H.P. and $ap,bq,cr$ are in G.P. then the value of the expression $\left(\frac{p}{r}+\frac{r}{p}\right)$ is equal to

If $a,b,c$ are real numbers and in A.P. and $a^2, b^2, c^2$ are in H.P; then which of the following option is correct:

If $x,y,z\in R$ satisfy the equation $2\log \left(x-z\right)-\log \left(x-2y+z\right)=\log \left(x+z\right)$, then $x,y \& z$  are in

If $2(y – a)$ is harmonic mean of terms $y – x \& y-z$, then $x – a, y – a \& z-a$  are in

If the values of first two terms of a harmonic progression series is $\frac{2}{5} \& \frac{12}{23}$ respectively, then the largest positive term of the progression is the

If the values of $4^{\text{th} }\& 8^{\text{th}}$ term of a H.P. is $\frac{3}{5} \& \frac{1}{3}$ respectively, then the value of $6^{th}$ term is:

If $a_1,a_2,a_3\dots \dots \dots \dots .. a_n$ are in HP then the value of the series $a_1 a_2+{a_2 a}_3+a_3 a_4\dots \dots .+{a_{n-1}a}_n=$

$H_1,H_2 \mathrm{a}\mathrm{r}\mathrm{e} 2 H.M{.}^{\prime }\text{s}$  inserted between $a,b$ then the value of the expression $\frac{H_1+H_2}{H_1{H}_2}$ is equal to

If three numbers $a, 8, b$ are forming an A.P., $a, 4, b$ are in G.P., $a, x, b$ are in H.P then the value of $x$ is equal to