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Alright I have this problem, Prove by induction $2^1+2^2+2^3+2^4+ \cdots +2^n=2(2^n-1)$

Now I've done this so far:

Base case $n=1$:

$$2^1 = 2$$

$$2(2^1-1)=2(2-1)=2(1)=2 .$$

Assume for $k$, prove for $k+1$:

$$ \begin{align*} 2^1+2^2+2^3+2^4+ \cdots +2^k+2^{k+1} & =2(2^k-1)+2^{k+1} \\ &=2^{k+1}+2^{k+1}-2 \end{align*} $$

Now the trouble I'm running into is that I don't know how to continue from here, I know that I need to show that somehow $2^{k+1}+2^{k+1}-2 = 2(2^{k+1}-1)$.

Is there something that I would be missing with the rules of exponents, or maybe I just made a mistake and I'm doing something wrong that I don't recognize?

Any help would be appreciated, thanks.

  • 0
    We have $2^{k+1} + 2^{k+1} = 2^{k+1} (1+1)$. I'll let you conclude.2011-12-03
  • 1
    $2^{k+1}+2^{k+1}=2\cdot 2^{k+1}$2011-12-03
  • 1
    You're almost finished, what's the problem? $2^{k+1}+2^{k+1}-2 = 2\cdot 2^{k+1}-2\cdot 1 = 2\cdot(2^{k+1}-1)$2011-12-03
  • 0
    Duplicate of this answer : http://math.stackexchange.com/questions/86838/the-sum-of-powers-of-2-between-20-and-2n/86844#868442011-12-04

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