Evaluate $$\lfloor\sqrt{1}\rfloor + \lfloor\sqrt{2}\rfloor + \lfloor\sqrt{3}\rfloor + .... + \lfloor\sqrt{19}\rfloor$$
This problem is asking me to add up the integer parts of the square roots of the first 19 positive integers. I notice that the integer part of a square root changes only when the argument is a perfect square. So, for example, $$\lfloor\sqrt{1}\rfloor = \lfloor\sqrt{2}\rfloor = \lfloor\sqrt{3}\rfloor = 1$$ Similarly, $$\lfloor\sqrt{4}\rfloor = \lfloor\sqrt{5}\rfloor = \lfloor\sqrt{6}\rfloor = \lfloor\sqrt{7}\rfloor = \lfloor\sqrt{8}\rfloor = 2$$ And so on, until $$\lfloor\sqrt{16}\rfloor = \lfloor\sqrt{17}\rfloor = \lfloor\sqrt{18}\rfloor = \lfloor\sqrt{19}\rfloor = 4$$ So, I can group the terms in the sum according to their values and count how many times each value appears. There are 3 terms with value 1, 5 terms with value 2, 7 terms with value 3, and 4 terms with value 4. Therefore, the sum is $$3 \times 1 + 5 \times 2 + 7 \times 3 + 4 \times 4$$ I can simplify this by multiplying and adding: $$3 + 10 + 21 + 16 = 50$$

