Question 7
Let $a_{1}, a_{2}, \ldots, a_{n}$ be positive real numbers, $n \gt 1$. Denote by $g_{n}$ their geometric mean, and by $A_{1}, A_{2}, \ldots, A_{n}$ the sequence of arithmetic means defined by $A_{k} = \frac{a_{1} + a_{2} + \cdots + a_{k}}{k}$, $k = 1, 2, \ldots, n$. Let $G_{n}$ be the geometric mean of $A_{1}, A_{2}, \ldots, A_{n}$. Prove the inequality
$$ n\sqrt[n]{\frac{G_{n}}{A_{n}}}+\frac{g_{n}}{G_{n}}\leq n+1 $$
and establish the cases of equality.
$$ n\sqrt[n]{\frac{G_{n}}{A_{n}}}+\frac{g_{n}}{G_{n}}\leq n+1 $$
and establish the cases of equality.