Generalization of $(a+b)^2\leq 2(a^2+b^2)$ We know that, $(a+b)^2\leq 2(a^2+b^2)$. Do we have anything similar for $$\left(\sum_{i=1}^N a_i\right)^2.$$
where $a_i\in \mathbb{R}\ \ \ \ \forall\ i\in \{1,\cdots,N\}$.
For $n=3$, we get
\begin{equation}
\begin{aligned}
(a_1+a_2+a_3)^2&\leq 2\left((a_1+a_2)^2+a_3^2\right)
\\&\leq 2\left(2(a_1^2+a_2^2)+a_3^2 \right).
\end{aligned}
\end{equation}
Do we have some sort of generalization?

It's C-S:
$$n(a_1^2+a_2^2+...+a_n^2)=$$
$$=(1^2+1^2+...+1^2)(a_1^2+a_2^2+...+a_n^2)\geq(a_1+a_2+...+a_n)^2.$$
