By Cauchy-Schwarz,
\[(a_1 + a_2 + \dots + a_n)(a_1^3 + a_2^3 + \dots + a_n^3) \ge (a_1^2 + a_2^2 + \dots + a_n^2)^2.\]Since $96 \cdot 216 = 144^2,$ we have equality in the Cauchy-Schwarz Inequality, which means
\[\frac{a_1^3}{a_1} = \frac{a_2^3}{a_2} = \dots = \frac{a_n^3}{a_n}.\]Then $a_1^2 = a_2^2 = \dots = a_n^2,$ so $a_1 = a_2 = \dots = a_n.$

From the given, $na_1 = 96$ and $na_1^2 = 144.$  Dividing these equations, we get $a_1 = \frac{3}{2},$ so $n = \boxed{64}.$
Question:
Let $(a_1, a_2, \dots, a_n)$ be a sequence of positive real numbers, such that
\[\sum_{i = 1}^n a_i = 96, \quad \sum_{i = 1}^n a_i^2 = 144, \quad \sum_{i = 1}^n a_i^3 = 216.\]Find the sum of all possible values of $n.$
Answer:
Let's think step by step.

user
64
Let $(a_1, a_2, \dots, a_n)$ be a sequence of positive real numbers, such that
\[\sum_{i = 1}^n a_i = 96, \quad \sum_{i = 1}^n a_i^2 = 144, \quad \sum_{i = 1}^n a_i^3 = 216.\]Find the sum of all possible values of $n.$
test