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How to see that $2^{n-1} + 2^{n-1} - 1 = 2^n - 1$?

Is there a rule about adding two powers of the same base I'm not aware of? I know that you can "add the exponents" if you are multiplying numbers of the same base, or "subtract" them if you are dividing.

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  • $\begingroup$ No, just take $2^{n-1}$ as a common factor. $\endgroup$
    – user207710
    Mar 14, 2015 at 16:49
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    $\begingroup$ $a+a=2\cdot a$ is true for all a; in this case, as the various answers have noted, you have $a=2^{n-1}$. $\endgroup$
    – Steven Stadnicki
    Mar 14, 2015 at 16:54

4 Answers 4

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The general rule is $$\underbrace{k^n+k^n+\cdots +k^n}_k=k\cdot k^n=k^1k^n=k^{1+n}$$

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$$2^{n-1} + 2^{n-1} - 1 = 2\cdot2^{n - 1}-1=2^{1+n-1}-1=2^n-1$$

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Simply $2^{n-1}+2^{n-1}=2\cdot 2^{n-1}=2^n$ which works for the base $2$ - for base three you'd need to add three times $3^{n-1}$.

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$2^{n-1} + 2^{n-1} - 1 = 2\cdot2^{n-1} - 1 = 2^n - 1$

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