Help on simple-looking ineqality

[phonic]

Dear all,

I come across to a simple-looking ineqality. But I cann't prove it for quite a long time. Could anybody give a hint? Thanks a lot!

$$2[(n-1) \sum_{j=1}^n r_j^2 -(n-2) r_n^2] \geq (\sum_{j=1}^n r_j)^2$$

where $$n\geq 2, \forall r_j \geq 0, j=1,2,\cdots,n$$.

 

[HallsofIvy]

Are you saying that you do not understand why (n-3)²r_n² should be non-negative?

 

[phonic]

Thanks for your sugestions. However, I still don't know how did you get to the simpification of $$(n-3)^2r_n^2$$.

According to your sugestions, I got the following proof:

$$2[(n-1)\sum_{j=1}^n r_j^2 -(n-2) r_n^2] =2[(n-1)\sum_{j=1}^{n-1} r_j^2 + r_n^2] \geq 2[(\sum_{j=1}^{n-1} r_j)^2 + r_n^2] \geq (\sum_{j=1}^{n-1} r_j)^2 + r_n^2 + 2r_n \sum_{j=1}^{n-1} r_j =(\sum_{j=1}^{n} r_j)^2$$