An identity via summation by parts

Question

In a text I am currently reading, the authors use through their computations the following identity : $$\sum_{i=1}^n \left( a_{i+1}-a_i-\frac{1}{n}\left(a_{i+n}-a_i\right)\right) v_i = \frac{1}{n}\sum_{i=1}^n \sum_{k=i+1}^{i+n}\left(a_i-a_k\right)\left(v_{i+1}-v_i\right) $$ where $v\in \mathbb{R}^n$ and, $$a_n=\sum_{i=1}^n b_i .$$ The justification given is by summation by parts : $$ \sum_{i=1}^n x_i y_i = y_n\left(\sum_{i=1}^n x_i\right)-\sum_{i=1}^{n-1}\left(\sum_{k=1}^i x_k \right)\left(y_{i+1}-y_i\right). $$ However I can't find the right path in applying the formula. Any help would be appreciated.