If $u_n = a + bn + c2^n$, what's the sum of the first $n$ terms?

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The \(n\)th term of a certain series is of the form \(a+bn+c2^n\), where \(a\), \(b\), and \(c\) are numbers. If the first three terms are \(2\), \(-1\), and \(3\), find the values of \(a\), \(b\), and \(c\), and the sum of the first \(n\) terms.