I have copied the first step of the solution below in order to save time and space, the problem, with full solution, can be found in the book *Putnam and Beyond*. 

Question:
Let $a\_1, a\_2, \ldots, a\_n, \ldots$ be a sequence of distinct positive integers. Prove that for any positive integer $n$,

$$a\_1^2 + a\_2^2 + \cdots + a\_n^2 \ge \tfrac{2n + 1}{3}(a\_1 + a\_2 + \cdots + a\_n).$$ 

Solution:
$$a\_{n+1}^2 \ge \tfrac{2n + 3}{3}a\_{n+1} + \tfrac{2}{3}(a\_1 + a\_2 + \cdots + a\_n) \ldots$$

Note:
 I understand how to make the jump from the question to the inductive step, namely replacing $n$ with $n+1$, but I don't see where the term $\tfrac{2}{3}(a\_1 + a\_2 + \cdots + a\_n)$ came from in the solution. Any help would be appreciated.