$ | \sum\limits_{j=1}^{n} a_j | ^2 $ + $ | \sum\limits_{j=1}^{n} (-1)^{j} a_j | ^2 \leq (n+2) \sum\limits_{j=1}^{n} a_j^2 $.
By simple expansion, $ LHS = 2\sum \limits_{j=1}^n {a_j^2} + 4\sum \limits_{j, k \in S }^n a_j a_k$, where $S$ is the set of all $ 1 \leq j < k \leq n$ where $j+k$ is even. Then $4\sum \limits_{j , k \in S }^n a_j a_k \leq \sum \limits_{s=1}^n {n_s a_s^2} $ where $n_s$ denotes the number of pairs $(j, k)$ in $S$ with $j = s$ or $ k = s$. One has $ n_s ≤ \left \lfloor{\frac{(n − 1)}{2}}\right \rfloor $, so substituting it into $\sum \limits_{s=1}^n {n_s a_s^2} $ obtains the solution.
Questions
- Why did the author assumed that $j + k$ must be even?
- An explanation of why his method had worked based on the $j + k$ condition.