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Find the sum, $f(n)$, of the first $n$ terms of the sequence: \begin{equation*} 0, 1, 1, 2, 2, 3, 3, \dots , p, p, p +1, p + 1, \dots \end{equation*}

Go on to prove that $f(a + b) - f(a - b) = ab$, where $a$ and $b$ are positive integers and $a > b$.