Is there any special relation between squared root of sum of squares and sum of the values themselves?

Question

I mean, is there any relation between $a_1 + a_2 + a_3 + ... + a_n$ and $\sqrt{a_1^2 + a_2^2 + a_3^2 + ... + a_n^2}$ ? This relation can be of any kind or any use.

Thank you all in advance.

Answer 1

Assuming that all the $a_i \geqslant 0$ then it is true that the quadratic mean is greater than the arithmetic mean: $$\left(\frac{1}{n}(a_1^2+a_2^2+\cdots+a_n^2)\right)^{\frac12} \geqslant \frac{1}{n}(a_1+a_2+\cdots+a_n)$$ or $$n(a_1^2+a_2^2+\cdots+a_n^2) \geqslant (a_1+a_2+\cdots+a_n)^2$$