Proving 2^n - 1 Formula for Positive Integers | Induction Question

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Homework Statement



Show that the statement holds for all positive integers n

2^0 + 2^1 + 2^2 + 2^3 + ... + 2^{n-1} = 2^n - 1

Homework Equations


The Attempt at a Solution



Assume:
2^0 + 2^1 + 2^2 + 2^3 + ... + 2^{k-1} = 2^k - 1 is true.

Then:
2^{k+1}-1 = 2^k(2) - 1 = 2(2^0 + 2^1 + 2^2 + 2^3 + ... + 2^{k-1})

Not sure what to do next :/

Show that:
2^k(2) - 1 = 2(2^0 + 2^1 + 2^2 + 2^3 + ... + 2^{k-1}) is true
 
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Hi zeion! :smile:

You have the right basic idea, but you're making it too complicated. :redface:

Do the obvious :wink:

just multiply both sides of the original equation by 2 (and then fiddle about with it). :smile:
 
Haven't I already proved it?

Since:
<br /> 2^0 + 2^1 + 2^2 + 2^3 + ... + 2^{k-1} = 2^k - 1 <br />

Then:
<br /> (2)(2^0 + 2^1 + 2^2 + 2^3 + ... + 2^{k-1}) = (2)2^k - 1 = 2^{k+1}-1<br />
 
No, it isn't … be careful! :wink:

(2)(2^0 + 2^1 + 2^2 + 2^3 + ... + 2^{k-1}) = (2)(2^k - 1) = \cdots\ ? :smile:
 
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