Question

If $a^2$ being the arithmetic mean (A.M.) of $xy$ and $\frac{x^2+y^2}{2}$ then prove that $a$ is A.M. of $x$ and $y$.

Answer 1

Given, $a^2$ being the arithmetic mean (A.M.) of $xy$ and $\frac{x^2+y^2}{2}$.

Then,

$a^2=\frac{xy+\frac{x^2+y^2}{2}}{2}$

or, $4a^2=(x+y{)}^2$

or,$2a=(x+y)$

or, $a=\frac{x+y}{2}$.

This gives $a$ is A.M. of $x$ and $y$.