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Add all these equations together. On the right-hand side, we get the sum \(1 + 2 + 3 + \cdots + n\text{.}\) We already know this can be simplified to \(\frac{n(n+1)}{2}\text{.}\) What happens on the left-hand side? We get

\begin{equation*} (a_1 - a_0) + (a_2 - a_1) + (a_3 - a_2) + \cdots (a_{n-1} - a_{n-2})+ (a_n - a_{n-1})\text{.} \end{equation*}

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