If b is the arithmetic mean between a and x; b is the geometric mean between ' a ' and y; ' b ' is the harmonic mean between a and z,(a, b, x, y, z0) then the value of x y z is

Question

If b is the arithmetic mean between a and x; b is the geometric mean between ' a ' and y; ' b ' is the harmonic mean between a and z,(a, b, x, y, z>0) then the value of x y z is (A) a 3 (B) b3 (C) (b3(2 a-b)/2 b-a) (D) (b3(2 b-a)/2 a-b)

Tardigrade Admin · Accepted Answer

2 b = x + a ....(1) b 2= ay ....(2) b =(2 az / a + z ) ....(3) x =2 b - a ; y =( b 2/ a ) text and (2/ b )=(1/ a )+(1/ z ) ⇒ z =( ab /2 a - b ) ∴ x y z=(2 b-a) (b2/a) ⋅ (a b/2 a-b)=(b3(2 b-a)/2 a-b)

Q. If $b$ is the arithmetic mean between $a$ and $x$ ; $b$ is the geometric mean between ' $a$ ' and $y$ ; ' $b$ ' is the harmonic mean between a and $z, (a, b, x, y, z > 0)$ then the value of $x yz$ is

$a^{3}$

$b^{3}$

$\frac{b ^{3} ( 2 a - b )}{2 b - a}$

$\frac{b ^{3} ( 2 b - a )}{2 a - b}$

$2 b = x + a$ ....(1)
$b^{2} = a y$ ....(2)
$b = \frac{2 a z}{a + z}$ ....(3)
$x = 2 b - a; y = \frac{b ^{2}}{a} and \frac{2}{b} = \frac{1}{a} + \frac{1}{z} \Rightarrow z = \frac{ab}{2 a - b}$
$∴ x yz = (2 b - a) \frac{b ^{2}}{a} \cdot \frac{ab}{2 a - b} = \frac{b ^{3} ( 2 b - a )}{2 a - b}$

Q. If b is the arithmetic mean between a and x; b is the geometric mean between ' a ' and y; ' b ' is the harmonic mean between a and z,(a,b,x,y,z>0) then the value of xyz is

a3

b3

2b−ab3(2a−b)​

2a−bb3(2b−a)​