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

Question

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.

Answer 1

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

Answer 2

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 $3^{n-1}$.

Answer 3

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

Answer 4

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