TITLE: Find the sum $\sum\limits_{k=1}^{2n} (-1)^{k} \cdot k^{2}$
QUESTION [5 upvotes]: How to find this sum?
$$\sum\limits_{k=1}^{2n} (-1)^{k} \cdot k^{2}$$

REPLY [8 votes]: As I'm a fan of proofs without words, here is my pictorial effort of why
$$\sum_{k=1}^{2n} (-1)^k k^2 = \sum_{i=1}^{2n} i = n(2n+1).$$