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.

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.