MathNet 2010 — Part 05 Question 200

Given $a_1 \ge 1$ and $a_{k+1} \ge a_k + 1$ for all $k = 1, 2, \dots, n$, show that
$$
a_1^3 + a_2^3 + \dots + a_n^3 \ge (a_1 + a_2 + \dots + a_n)^2.
$$