## anonymous 3 years ago Find minimum of : \sum_{k=1}^{n}a_{k}^{2}+\left(\sum_{k=1}^n a_k\right)^2. My teacher told me that I shoul use Cauchy-Schwarz inequality. Any tips?

1. anonymous

$\sum_{k=1}^{n}a_{k}^{2}+\left(\sum_{k=1}^n a_k\right)^2$ $\left(\sum_{k=1}^n a\right)^2=\sum_{k=1}^n a^3$ So your question boils down to $\sum_{k=1}^n a^2+ a^3=\sum_{k=1}^n a^2(a+1)$

2. anonymous
3. anonymous

The link is the ()^2=(^3) proof

4. anonymous

Although that's not Cauchy Schwartz.

5. anonymous

I didn't realise that task has another assumption :$\sum_{k=1}^{n}p_ka_k=1$

6. anonymous

Sorry, I forgot that a1 does not necessarily equal 1, a2=2 etc. forget what I've told you so far.

7. anonymous

http://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality#Statement_of_the_inequality you want the final formula of this section

8. anonymous

Or do you think a1=1, a2=2 etc in this problem?