Solution to Prove that \sum_{i=0}^{n} 2^{i} = 2^{n + 1} - 1 Use mathematical induction for this … - Sikademy
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Prove that \sum_{i=0}^{n} 2^{i} = 2^{n + 1} - 1 Use mathematical induction for this proof and discuss/explain each step.

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Let us prove that \sum\limits_{i=0}^{n} 2^{i} = 2^{n + 1} - 1 using mathematical induction.


For n=1, we have that the left side is 2^0+2^1=3, and the right side is 2^2-1=3. Therefore, for n=1 the equality is true.


Suppose that the equality is true for n=k, that is \sum\limits_{i=0}^{k} 2^{i} = 2^{k + 1} - 1.

Let us prove for n=k+1.


It follows that


\sum\limits_{i=0}^{k+1} 2^{i} =\sum\limits_{i=0}^{k} 2^{i}+2^{k+1} = 2^{k + 1}-1+ 2^{k + 1} =2\cdot 2^{k + 1} - 1=2^{k+2}-1.


We conclude that by principle of mathematical induction the statement is true for all natural numbers n.


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