Harmonic Mean

Q.

Find the sum of all odd numbers between $0$ and $50$.

Q. If $9$ arithmetic means and harmonic means are inserted between $2$ and $3$, then the value of $A+\frac{6}{H}$ is
(where $A$ is any of the A.M.'s and H the corresponding H.M.)

Q. The value of $x+y+zis15ifa,x,y,z,b$ are in A.P. while the value of
$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}is\frac{5}{3}ifa,x,y,z,b$ are in H. P. Then the value of and b are

1. 2 and 8
2. 1 and 9
3. 3 and 7
4. None

Q. If $a,b,c$ are in $H.P.,$ then $\frac{a}{b+c},\frac{b}{c+a},\frac{c}{a+b}$ are in

1. $A.P.$
2. $G.P.$
3. $H.P.$
4. None

Q. The harmonic mean of $\frac{a}{1-ab}$ and $\frac{a}{1+ab}$ is
[MP PET 1996; Pb. CET 2001]

1. $\frac{a}{\sqrt{1-a^2{b}^2}}$
2. $\frac{a}{1-a^2{b}^2}$
3. a
4. $\frac{1}{1-a^2{b}^2}$

Q. If the A.M. between a and b is m times their H.M. then a:b =

1. $\sqrt{m}+\sqrt{m-1}:\sqrt{m}-\sqrt{m-1}$
2. $\sqrt{m}-\sqrt{m-1}:\sqrt{m+1}+\sqrt{m-1}$
3. $\sqrt{m}+\sqrt{m+1}:\sqrt{m}-\sqrt{m+1}$
4. None of these

Q. If the harmonic mean between a and b be H, then the value of $\frac{1}{H-a}+\frac{1}{H-b}$ is

1. a+b
2. ab
3. $\frac{1}{a}+\frac{1}{b}$
4. $\frac{1}{a}-\frac{1}{b}$

Q. Let $A=a^{2}b+a{b}^2-a^{2}c-a{c}^2,$ $B=b^{2}c+b{c}^2-a^{2}b-a{b}^2$ and $C=a^{2}c+a{c}^2-b^{2}c-b{c}^2$, where $a>b>c>0$. If the equation $A{x}^2+Bx+C=0$ has equal roots, then $a,b,c$ are in

1. A.P.
2. G.P.
3. H.P.
4. A.G.P.

Q. If $H_1,H_2,\dots ,H_{20}$ be $20$ harmonic means between $2$ and $3$, then the value of $\frac{H_1+2}{H_1-2}+\frac{H_{20}+3}{H_{20}-3}$ is

1. $20$
2. $21$
3. $40$
4. $41$

Q. If $H_1,H_2,\dots \dots ,H_{20}$ be 20 harmonic means between $2$ and $3$, then the value of $\frac{H_1+2}{H_1-2}+\frac{H_{20}+3}{H_{20}-3}$ is

1. $40$
2. $21$
3. $20$
4. $41$