<s>[INST] Why does $2^{n+1} + 2^{n+1} = 2^{n+2}$? Simple question, why does:
$2^{n+1} + 2^{n+1} = 2^{n+2}$ ?
Furthermore, why does this only work for powers of 2?
Thanks.
 [/INST] $$2^{n+1} + 2^{n+1} = 2\cdot 2^{n+1} = 2^1 \cdot 2^{n+1} = 2^{n+1 + 1} = 2^{n+2}$$
We use the fact that $a^n\cdot a^m = a^{n+m}$.
Added: For larger bases, say we have an integer base $a$, then 
$$\underbrace{a^{n+1}+a^{n+1} + \cdots + a^{n+1}}_{\large a \text{ terms } }= a\cdot a^{n+1} = a^{n+2}$$
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