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Mathematics
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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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Proof of the sum of the geometric sequence.
Very nice. Did you find this by yourself?

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Other answers:

Yep
total permutations possible when # of states are 2,3,4,... x awesome stuff !
I have another proof for this, but yours is pretty cool. :)
superb
Nice, I did it a slightly less ingenious way, but I do like your proof!
this is amazing work !
tsk. math lovers.....

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