How does $2^{k+1} = 2 \times 2^k$?
I ask only because my textbook infers this in an example. Where should I go to learn more about this?
I'm trying to learn mathematics by Induction but my knowledge of simplifying algebraic equations is crippling me.
Thanks.
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$\begingroup$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 $$x^k \times x = \underbrace{x \times x \times \cdots \times x}_{k \textrm{ times}} \times x = \underbrace{x \times x \times \cdots \times x}_{k+1 \textrm{ times}}.$$
$\endgroup$ 1 $\begingroup$$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 may help.
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