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Math Help - greatest integers

  1. #1
    rcs
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    greatest integers

    the [x] means the greatest integer less than or equal to x.

    e.g., [5.7] = 7, [pi] = 3, [4] = 4

    greatest integers-functions-sum.jpg
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  2. #2
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    Quote Originally Posted by rcs View Post
    the [x] means the greatest integer less than or equal to x.

    e.g., [5.7] = 7, [pi] = 3, [4] = 4

    Calculate the value of the sum

    \lfloor\sqrt1\rfloor + \lfloor\sqrt2\rfloor + \lfloor\sqrt3\rfloor + \lfloor\sqrt4\rfloor + \ldots + \lfloor\sqrt{48}\rfloor + \lfloor\sqrt{49}\rfloor + \lfloor\sqrt{50}\rfloor.
    [Actually \lfloor5.7\rfloor is 5, not 7.]

    The first three terms in that long sum will all be equal to 1 (because \sqrt2 and \sqrt3 both lie between 1 and 2). The next batch of terms, from \lfloor\sqrt4\rfloor to \lfloor\sqrt8\rfloor will all be equal to 2. Count how many of these terms there are. Then do the same for the batches of terms that are equal to 3, 4, 5, 6. Finally, the last two terms are equal to 7.

    Now add all those numbers to get the answer.
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  3. #3
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    Hello, rcs!

    Opalg is absolutely correct!
    Here's some trivia that shouldl help . . .


    \text{Evaluate: }\;[\sqrt{1}\,] + [\sqrt{2}\,] + [\sqrt{3}\,] + \hdots + [\sqrt{50}\,]

    . . \text{where }[x]\text{ is the greatest integer function.}

    We note that squares differ by consecutive odd numbers.

    . . \begin{array}{cccccccccccc}<br />
\text{Squares:} & 1 && 4 && 9 && 16 && 25 && \hdots \\<br />
\text{Difference:} && 3 && 5 && 7 && 9 && \hdots \end{array}


    As Opalg explained:

    . . [\sqrt{1}\,],\;[\sqrt{2}\,],\;|\sqrt{3}\,] are all equal to 1.

    . . [\sqrt{4}\,],\;[\sqrt{5}\,],\;[\sqrt{6}\,],\;[\sqrt{7}\,],\;[\sqrt{8}\,] are all equal to 2.

    . . [\sqrt{9}\,]\;[\sqrt{10}\,],\; [\sqrt{11}\,]\;\hdots\;[\sqrt{15}\,] are all equal to 3.

    . . and so on.


    So we have:

    . . \begin{array}{ccc}<br />
[\sqrt{1}\,] + [\sqrt{2}\,] + [\sqrt{3}\,] &=& 3(1) \\<br /> <br />
[\sqrt{4}\,] + [\sqrt{5}\,] + \hdots + [\sqrt{8}\,] &=& 5(2) \\<br /> <br />
[\sqrt{9}\,] + [\sqrt{10}\,] + \hdots + [\sqrt{15}\,] &=& 7(3) \\<br /> <br />
[\sqrt{16}\,]+[\sqrt{17}\,] + \hdots + [\sqrt{24}\,] &=& 9(4) \\<br /> <br />
[\sqrt{25}\,] + [\sqrt{26}\,] + \hdots [\sqrt{35}\,] &=& 11(5) \\<br /> <br />
[\sqrt{36}\,] + [\sqrt{37}\,] + \hdots [\sqrt{48}\,] &=& 13(6) \\<br /> <br />
[\sqrt{49}\,] + [\sqrt{50}\,] &=& 2(7) \\ <br /> <br />
& & --- \\<br /> <br />
& & \text{Total?}<br /> <br />
\end{array}
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