# How does $2^{k+1} = 2 \times 2^k$?

I'm trying to learn mathematics by Induction but my knowledge of simplifying algebraic equations is crippling me.

Thanks.

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When you don't understand generalized formulas, expressions, etc., try making sense of them by using concrete examples. For instance $2 \cdot 2^2 = 2 \cdot 4 = 8 = 2^3 = 2^{2+1}$, etc. Then you can grasp what's happening or why the generalization makes sense. Also, Stefan's link to exponentiation below may help. – amWhy Oct 30 '12 at 20:09
Someone should probably mention that the formula you are asking about is usually taken to be the formal definition of integer exponentiation (it is a definition by recursion.) So if you are having trouble finding a formal proof of it, that's why. – Trevor Wilson Oct 30 '12 at 20:56

By the rules of exponentiation,

$x^{k} \times x = x^{k+1}$.

If $k$ is an integer, $x^k = \underbrace{x \times x \times \cdots \times x}_{k \textrm{ times}}.$

So $$x^k \times x = \underbrace{x \times x \times \cdots \times x}_{k \textrm{ times}} \times x = \underbrace{x \times x \times \cdots \times x}_{k+1 \textrm{ times}}.$$

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unless $x$ is a Grassmann number ;) – Valentin Oct 30 '12 at 21:20

$2^{k+1}$ is $2$ multiplied with itself k+1 times. $2\cdot2^k$ is $2$ multiplied $k$ times with itself and an additional $2$ makes it multiplied $k+1$ times with itself.

Also a look at http://en.wikipedia.org/wiki/Exponentiation may help.

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Nevermind, I'm just having a hard time wrapping my head around it. I'll just have to accept that it's true. – Unknown Oct 30 '12 at 20:03
You dont have to accept it, accepting is like believing, you can't be sure about it. If you really want to learn maths, you should question everything possible, that way you can really get behind a subject and understand what you are doing. Take a look at the wiki article, maybe it helps! – Stefan Oct 30 '12 at 20:07
Oh, it makes sense. I was confused, but it is quite simple. Since multiplying 2 by $2^k$ is just like increasing k by 1. Very easy stuff. Thanks. – Unknown Oct 30 '12 at 20:16