The sum of the first n terms of the sequence 1, (1+2), (1+2+2^2), …, (1+2+2^2+…+2^n-1) in terms of n is:

1972 AMC 12 Problem 19

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19.

The sum of the first

n

terms of the sequence

\begin{gathered} 1,\ (1+2),\ (1+2+2^2),\\ \ldots,\ (1+2+2^2+\cdots+2^{n-1}) \end{gathered}

in terms of

n

is:

2^n

2^n-n

2^{n+1}-n

2^{n+1}-n-2

n\cdot2^n

The

k

-th parenthesized sum is

2^k-1

Sum

2^1-1,

2^2-1,

\ldots,

and

2^n-1

The

k

-th term is the geometric sum

2^k-1.

Therefore the desired total is

\begin{aligned} \sum_{k=1}^n(2^k-1) &=(2^{n+1}-2)-n\\ &=2^{n+1}-n-2. \end{aligned}

Therefore, the correct answer is D.