Obtaining An Expression For the nth Term of a Sequence Given An Expression For The Sum Of n Terms

Suppose we are given an expression  
\[S_n\]
  for the sum of the first  
\[n\]
  terms of a series. The  
\[n\]
th term is given by  
\[a_n=S)n-S_{n-1}\]
.
Example:  
\[S_n==n^3+2n-2\]
.
\[S_{n-1}=(n-1)^3+2(n-1)-2=n^3-3n^2+3n-1+2m-2-2=n^3-3n^2+5n-5\]

\[\begin{equation} \begin{aligned} a_n &=S_n-S_{n-1} \\ &= (n^3+2n-2)-(n^3-3n^2+5n-5) \\ &= 3n^2-3n+3(n^3+2n-2) \end{aligned} \end{equation}\]

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