# Is there any relationship between the geometric mean and the arithmetic mean?

Is there, at all any relationship between the geometric mean of a set and the arithmetic mean of a set? If I knew one, and I knew the number of terms in the set, how could I calculate the other?

In fact, given any two positive numbers $$a$$ and $$g$$ with $$a \ge g$$, there will always be two numbers whose arithmetic and geometric means are $$a$$ and $$g$$ respectively. Indeed, to find those two numbers, it suffices to solve the equation $$x(2a-x)=g^2$$. This equation can be rearranged as $$-x^2+2ax-g^2=0$$. Using the quadratic formula, the two solutions for $$x$$ are $$-\frac{1}{2}(-2a \pm \sqrt{4a^2-4g^2})$$. Those two solutions will then have AM $$a$$ and GM $$g$$.
• Because if $(x+y)/2=a$ and $\sqrt{xy}=g$, then $y=2a-x$ and $x(2a-x)=xy=g^2$. Feb 8 '20 at 23:13