1972 AHSME Problem 19

The sum of the first n terms of the sequence 1,~(1+2),~(1+2+2^2),~\ldots ~(1+2+2^2+\ldots +2^{n-1}) in terms of n is

\textbf{(A) }2^n\qquad \textbf{(B) }2^n-n\qquad \textbf{(C) }2^{n+1}-n\qquad \textbf{(D) }2^{n+1}-n-2\qquad \textbf{(E) }n\cdot 2^n

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.