Question

Let \[\sum_{k=1}^{n}p_ka_k=1\]. Find minimum of \[\sum_{k=1}^{n}a_k^2+\left (\sum_{k=1}^{n}a_k \right )^2\]. I've got answer but I don't undestrand proof. It begins with : \[1=\sum_{k=1}^{n}(p_k- \alpha )a_k + \alpha \sum_{k=1}^{n}a_k

\left ( \left ( \sum_{k=1}^{n}(p_k- \alpha )^2 \right ) + \alpha ^2 \right )
\left ( \sum_{k=1}^{n}a_k^2 + \left ( \sum_{k=1}^{n}a_k \right )^2 \right ) \geq 1 \]
( Using Cauchy-Schwarz inequality).

Answer 1

I've tried to do it but i'm stuck at \[1 \le\left( \sum_{k=1}^{n}(p_k- \alpha )^2 + \alpha ^2 \right )
\right) \sum_{k=1}^{n}a_k^2 +2 \alpha \sum_{k=1}^{n} (p_k- \alpha )a_k\sum_{k=1}^{n}a_k\]