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What is the sum of the following sequence

$$\begin{align*} (2^1 - 1) &+ \Big((2^1 - 1) + (2^2 - 1)\Big)\\ &+ \Big((2^1 - 1) + (2^2 - 1) + (2^3 - 1) \Big)+\ldots\\ &+\Big( (2^1 - 1)+(2^2 - 1)+(2^3 - 1)+\ldots+(2^n - 1)\Big) \end{align*}$$

I tried to solve this. I reduced the equation into the following equation

$$n(2^1) + (n-1)\cdot2^2 + (n-2)\cdot2^3 +\ldots$$

but im not able to solve it further. Can any one help me solve this equation out. and btw its not a Home work problem. This equation is derived from some puzzle.

Thanks in advance

  • 1
    Note that your formula $n(2^1) + (n-1)\cdot2^2 + (n-2)\cdot2^3 +\ldots$ is not correct, for example, when n=2 it does not yield 5.2012-09-07

3 Answers 3