1. ## greatest integers

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

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

2. Originally Posted by rcs
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

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

The first three terms in that long sum will all be equal to 1 (because $\displaystyle \sqrt2$ and $\displaystyle \sqrt3$ both lie between 1 and 2). The next batch of terms, from $\displaystyle \lfloor\sqrt4\rfloor$ to $\displaystyle \lfloor\sqrt8\rfloor$ 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.

3. Hello, rcs!

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

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

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

We note that squares differ by consecutive odd numbers.

. . $\displaystyle \begin{array}{cccccccccccc} \text{Squares:} & 1 && 4 && 9 && 16 && 25 && \hdots \\ \text{Difference:} && 3 && 5 && 7 && 9 && \hdots \end{array}$

As Opalg explained:

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

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

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

. . and so on.

So we have:

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