Knowing the arithmetic, geometric, and harmonic mean of a set, can I find the number of terms in the set? If not, what more information do I need?

Question

Consider a set $a$ with $t$ terms. Knowing the arithmetic mean, the geometric mean, and the harmonic mean of $a$, could I somehow solve for $t$? If not, what more information would I need to know?

Answer 1

For a nontrivial example, let $$(x_1, x_2, x_3) = (2, 1, \tfrac{1}{2}),$$ so that the arithmetic, geometric, and harmonic means are $$(\tfrac{7}{6}, 1, \tfrac{6}{7}).$$ Then consider $$(y_1, y_2) = (y, \tfrac{1}{y})$$ for some $y > 1$. Their arithmetic, geometric, and harmonic means are $$\left(\frac{y+y^{-1}}{2}, 1, \frac{2}{y+y^{-1}}\right),$$ hence $y$ satisfies $$\frac{7}{6} = \frac{y + y^{-1}}{2}$$ or $$y = \frac{7 + \sqrt{13}}{6}.$$ This furnishes a counterexample in which given the three means, there are two distinct sets of different sizes that give the same means.