Let $ \, a_{0}, a_{1}, a_{2},\ldots\,$ be a sequence of positive real numbers satisfying $ \, a_{i-1}a_{i+1}\leq a_{i}^{2}\,$ for $ i = 1,2,3,\ldots\; .$ (Such a sequence is said to be log concave.) Show that for each $ \, n > 1,$ \[ \frac{a_{0}+\cdots+a_{n}}{n+1}\cdot\frac{a_{1}+\cdots+a_{n-1}}{n-1}\geq\frac{a_{0}+\cdots+a_{n-1}}{n}\cdot\frac{a_{1}+\cdots+a_{n}}{n}.\]