Question 142
The sequence $a_1, a_2, \dots$ of positive real numbers satisfies
$$
a_{k+1} \ge \frac{ka_k}{a_k^2 + k - 1}
$$
for every positive integer $k$.
Prove that $a_1 + a_2 + \dots + a_n \ge n$ for every $n \ge 2$.
$$
a_{k+1} \ge \frac{ka_k}{a_k^2 + k - 1}
$$
for every positive integer $k$.
Prove that $a_1 + a_2 + \dots + a_n \ge n$ for every $n \ge 2$.