Let x be the arithmetic mean and y, z be the two geometric means between any two positive numbers. The value of (y3+ z3/xyz) is

Question

Let x be the arithmetic mean and y, z be the two geometric means between any two positive numbers. The value of (y3+ z3/xyz) is (A) 1 (B) 2 (C) 3 (D) 4

Tardigrade Admin · Accepted Answer

Let the two numbers be a and b. x = A.M. = (a+b/2) a, y = ar, z= ar2, b= ar3 are in G. P. Now, (y3+z3/xyz) =(a3r3+a3r6/((a+b))2 ⋅ a2 r3) = (2a(1+r3)/a+ar3) = 2 .

Q. Let $x$ be the arithmetic mean and $y, z$ be the two geometric means between any two positive numbers. The value of $\frac{y ^{3} + z ^{3}}{x yz}$ is

$1$

$2$

$3$

$4$

Let the two numbers be $a$ and $b$ .
$x = A . M . = \frac{a + b}{2}$
$a, y = a r, z = a r^{2}, b = a r^{3}$ are in $G . P$ .
Now, $\frac{y ^{3} + z ^{3}}{x yz}$
$= \frac{a ^{3} r ^{3} + a ^{3} r ^{6}}{\frac{( a + b )}{2} \cdot a ^{2} r ^{3}}$
$= \frac{2 a ( 1 + r ^{3} )}{a + a r ^{3}} = 2$ .

Q. Let x be the arithmetic mean and y,z be the two geometric means between any two positive numbers. The value of xyzy3+z3​ is

1

2

3

4

Let the two numbers be a and b. x=A.M.=2a+b​ a,y=ar,z=ar2,b=ar3 are in G.P. Now, xyzy3+z3​ =2(a+b)​⋅a2r3a3r3+a3r6​ =a+ar32a(1+r3)​=2.

Q. Let $x$ be the arithmetic mean and $y, z$ be the two geometric means between any two positive numbers. The value of $\frac{y ^{3} + z ^{3}}{x yz}$ is

$1$

$2$

$3$

$4$

Let the two numbers be $a$ and $b$ .
$x = A . M . = \frac{a + b}{2}$
$a, y = a r, z = a r^{2}, b = a r^{3}$ are in $G . P$ .
Now, $\frac{y ^{3} + z ^{3}}{x yz}$
$= \frac{a ^{3} r ^{3} + a ^{3} r ^{6}}{\frac{( a + b )}{2} \cdot a ^{2} r ^{3}}$
$= \frac{2 a ( 1 + r ^{3} )}{a + a r ^{3}} = 2$ .