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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?

Mathematics
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\[ \sum_{k=1}^{n}a_{k}^{2}+\left(\sum_{k=1}^n a_k\right)^2\] \[\left(\sum_{k=1}^n a\right)^2=\sum_{k=1}^n a^3\] So your question boils down to \[\sum_{k=1}^n a^2+ a^3=\sum_{k=1}^n a^2(a+1)\]
http://users.tru.eastlink.ca/~brsears/math/oldprob.htm#s32
The link is the ()^2=(^3) proof

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Other answers:

Although that's not Cauchy Schwartz.
I didn't realise that task has another assumption :\[\sum_{k=1}^{n}p_ka_k=1\]
Sorry, I forgot that a1 does not necessarily equal 1, a2=2 etc. forget what I've told you so far.
http://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality#Statement_of_the_inequality you want the final formula of this section
Or do you think a1=1, a2=2 etc in this problem?

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