Inequality Introductory Problem 2

Problem

Show that .

Solutions

First Solution

Working backwards from the next inequality we solve the origninal one:

$\begin{eqnarray*} \left(\sum_{k=1}^{n-1} (a_k-a_{k+1})^2\right) + (a_n-a_1)^2&\ge& 0\\ 2\cdot \left(\sum_{k=1}^n a_k^2\right) - 2\left(\left(\sum_{k=1}^{n-1} a_ka_{k+1}\right) +a_na_1\right)&\ge& 0\\ 2\cdot \left(\sum_{k=1}^n a_k^2\right) &\ge& 2\left(\left(\sum_{k=1}^{n-1} a_ka_{k+1}\right) +a_na_1\right)\\ 2\cdot \sum_{k=1}^n a_k^2 &\ge& 2(a_1a_2+a_2a_3+\cdots+a_{n-1}a_n+a_na_1)\\ \sum_{k=1}^n a_k^2 &\ge& (a_1a_2+a_2a_3+\cdots+a_{n-1}a_n+a_na_1) \end{eqnarray*}$ (Error compiling LaTeX. Unknown error_msg)