(From the book The Cauchy-Schwarz Master Class by J. Michael Steele, Exercise 2.10) For nonnegative \(a_1, a_2, \ldots, a_n\) and \(n \ge 2\) $$ a_n \bigg(\frac{a_1+ a_2+ \ldots + a_{n-1}}{n-1} \bigg)^{n-1} \le \bigg(\frac{a_1+ a_2+ \ldots + a_n}{n} \bigg)^n $$ Proof. Let \(y_i = \frac{a_1+ a_2+ \ldots + a_i}{i}\) for \(i \ge 2 \). Then the …