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This was taken from Brilliant Floor Function | Brilliant Math & Science Wiki \(\displaystyle S = \lfloor{\sqrt{1}}\rfloor + \lfloor{\sqrt{2}}\rfloor + ... + \lfloor{\sqrt{1988}}\rfloor\) Find the value of S. As usual, put answers in spoiler tags and show your work. This might be on the simpler side, but floor and ceiling functions don't get a lot of attention (at least in my experience)
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Spoiler \(\displaystyle S = \lfloor{\sqrt{1}}\rfloor + \lfloor{\sqrt{2}}\rfloor + ... + \lfloor{\sqrt{1988}}\rfloor\) \(\displaystyle = (1988 + 1)\lfloor\sqrt{1988}\rfloor + \frac{1}{3}\left(-(\lfloor\sqrt{1988}\rfloor+ 1)^3 + \frac{3}{2}(\lfloor\sqrt{1988}\rfloor + 1)^2 + \frac{1}{2}(-\lfloor\sqrt{1988}\rfloor - 1)\right)\) \(\displaystyle =58146\)
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Spoiler \(\displaystyle \sum _{k=1}^n \left\lfloor \sqrt{k}\right\rfloor =m\left(n-\frac{(2m+5)(m-1)}{6}\right)\) where \(\displaystyle m=\lfloor \sqrt n\rfloor \)
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MacstersUndead said: This was taken from Brilliant Floor Function | Brilliant Math & Science Wiki \(\displaystyle S = \lfloor{\sqrt{1}}\rfloor + \lfloor{\sqrt{2}}\rfloor + ... + \lfloor{\sqrt{1988}}\rfloor\) Find the value of S. As usual, put answers in spoiler tags and show your work. This might be on the simpler side, but floor and ceiling functions don't get a lot of attention (at least in my experience) Click to expand... Puzzles like this, on a platform like MHF seem to be oblivious to the likelihood that anyone reading it is sitting in front of a computer. This took more time to work out a feasible hand strategy than to hack the 1 2 or 3 lines of code needed to get the machine to check the answer! It would of have taken a little longer to frame the question to get WA to check it. The guys and gals at Bletchley would of course just have lashed together a room sized electro-mechanical special purpose computer to check the answer, while those at LSE would have tried to check it using an hydraulic analog simulator.
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Spoiler You're right, and it was the way I did it (after writing a closed form based on an observation simplifying the first 16 terms)
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Spoiler \(\displaystyle \sqrt1= 1,\ \sqrt4=2,\ \sqrt9 = 3\ \sqrt{16}=4,\ ...\ ,\sqrt{1934} = 44\) The rest, if I am correct, is just simple calculations.[spoiler/]
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