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How many different terms in this sequence? $u_k=\left\lfloor \frac{k^2}{2013}\right\rfloor$, $k=1,2,3...2013$

Thanks so much

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2 Answers 2

up vote 5 down vote accepted

Hint: when $k$ is small enough, $(k+1)^2-k^2 \lt 2013$ and the floor will have you hit every number. When $k$ gets larger, the change will be larger than $2103$ and all the values will be distinct. A little thought about the transition and you are there.

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Each $k$ is different, so the answer should be $2013$ right?

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No, because of the floor. $\lfloor \frac {1^2}{2013} \rfloor = \lfloor \frac {2^2}{2013}\rfloor =\lfloor \frac {3^2}{2013}\rfloor =0$ –  Ross Millikan Nov 14 '12 at 14:44
    
Not right, with k=1,2,...,44 then $u_k=0$ –  tangkhaihanh Nov 14 '12 at 14:44

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