Question

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

• A. $2$
• B. $3$
• C. $\frac{1}{2}$
• D. $\frac{3}{2}$

Answer 1

The correct option is A $2$

Let a and b the two numbers
Lets consider $a=p^3,b=q^3$
$x=p^{2}q$
$y=p{q}^2$
$(x{)}^3+(y{)}^3=\left(pq{)}^3\right[p^3+q^3]$
$=ab(a+b)$
$z=(a+b)/2$
$2z=a+b$
So the expression for $(x{)}^3+(y{)}^3=2zab$
Now $ab=p^3{q}^3=\left(p^{2}q\right)\left(p{q}^2\right)=xy$
Thus $x^3+y^3=2xyz$
$(x^3+y^3)/xyz=2$