Question 112
Given a sequence $a_{1}, a_{2}, a_{3}, \ldots$ of non-negative real numbers satisfying the conditions
(1) $a_{n} + a_{2n} \geq 3n$
(2) $a_{n+1} + n \leq 2 \sqrt{a_{n} \cdot (n+1)}$
for all indices $n = 1, 2, \ldots$
a. Prove that the inequality $a_{n} \geq n$ holds for every $n \in \mathbb{N}$.
b. Give an example of such a sequence.
(1) $a_{n} + a_{2n} \geq 3n$
(2) $a_{n+1} + n \leq 2 \sqrt{a_{n} \cdot (n+1)}$
for all indices $n = 1, 2, \ldots$
a. Prove that the inequality $a_{n} \geq n$ holds for every $n \in \mathbb{N}$.
b. Give an example of such a sequence.