<s>[INST] Why is it intuitively "obvious" that: $2^0+2^1+2^2+...+2^n=2^{n+1}-1$ Why is it "obvious" that: $$2^0+2^1+2^2+...+2^n=2^{n+1}-1$$
And how about: $$2^{-1}+2^{-2}+2^{-3}+...+2^{-n}=1-2^{-n}$$
I am looking for some intuition about these facts.
Is there a similar pattern when the base is $3$?
 [/INST] Adding another answer to the very good proofs below : you are already familiar with this concept each time you use numbers.
$999_{10} + 1 = 1000_{10}$ is the same than $(9 \times 10^2)+(9 \times 10^1)+(9 \times 10^0) +1 = 10^3$.
If you set $b=9_{10}$ you wrote $(b-1) \times b^2+ (b-1) \times b^1+(b-1) \times b^0 +1 = b^3 = (b-1) ( b^2+b^1+b^0 )$
In base 2, it is the same :
$111_2 = (1 \times 2^2)+(1 \times 2^1)+(1 \times 2^0) +1 = 2^3$.
If you set $b=2_{10}$ you wrote $(b-1) \times b^2+ (b-1) \times b^1+(b-1) \times b^0 +1 = b^3 = b^2+b^1+b^0+1$
Not a proof, just a recall of a knowledge already acquired ...
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