How to prove the sum of squares larger than 1/n without induction?

Question

known that:
$1\geq R_1 \geq R_2 \geq \dots \geq R_n \geq 0$ and $\sum_{i=1}^n R_i=1$
To prove:
$\sum_{i=1}^n R_i^2 \geq \frac{1}{n}$

Using induction, the problem can be easily proved. I'd like to know is there any other ways can prove this?

Answer 1

It is the inequality arithmetic mean-quadratic mean $$\frac1n=\frac{\sum_i R_i}n\le \biggl(\frac{\sum_i R_i^2}n\biggr)^{\!\tfrac12}\iff\frac1{n^2}\le\frac{\sum_i R_i^2}n\iff \frac1n\le\sum_i R_i^2.$$