linear algebra - How are these cross-product summations equivalent?

Trying to determine how the $X_{i+1}$ is no longer applicable by changing summation bounds:

$$\sum_{i=0}^{n-1} (X_{i} + X_{i+1})(Y_{i+1} - Y_{i}) = \sum_{i=1}^{n} X_{i}(Y_{i+1} - Y_{i - 1})$$

Can somebody explain algebraically how this is derived from the bounds?

The source is found here, scroll down to "Polygons, 2D Polygons".

Answer

There was a previously deleted answer which was almost correct, which I have fixed and reproduced below:

\begin{eqnarray}
&&\sum_{i=0}^{n-1} (X_{i} + X_{i+1})(Y_{i+1} - Y_{i})\\
&=&\sum_{i=0}^{n-1}X_{i}(Y_{i+1} - Y_{i})+\sum_{i=0}^{n-1} X_{i+1}(Y_{i+1} - Y_{i})\\
&=&\sum_{i=0}^{n-1}X_{i}(Y_{i+1} - Y_{i})+\sum_{i=1}^{n} X_{i}(Y_{i} - Y_{i-1})\\
&=&\sum_{i=1}^{n}X_{i}(Y_{i+1} - Y_{i})+\sum_{i=1}^{n} X_{i}(Y_{i} - Y_{i-1})\\
&=&\sum_{i=1}^{n}X_{i}(Y_{i+1} - Y_{i-1})
\end{eqnarray}
The third equality follows since $X_0(Y_1-Y_0)=X_n(Y_{n+1}-Y_n)$ (as the indices are modulo $n$), so removing the $i=0$ term and adding in the $i=n$ term does not change the sum.