Question

The symbolism $\lfloor x \rfloor$ denotes the largest integer not exceeding $x.$ For example, $\lfloor 3 \rfloor = 3,$ and $\lfloor 9/2 \rfloor = 4.$ Compute $\begin{gathered} \lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor \\ {}+ \cdots + \lfloor \sqrt{16} \rfloor. \end{gathered}$

$35$

$38$

$40$

$42$

$136$

Answer: B

Concepts:floor and ceiling functions perfect square summation

Difficulty rating: 1170

Solution:

The value is $1$ for $n=1,2,3;$ it is $2$ for $n=4,\ldots,8;$ it is $3$ for $n=9,\ldots,15;$ and it is $4$ for $n=16.$ The sum is $3\cdot 1 + 5\cdot 2 + 7\cdot 3 + 1\cdot 4 = 38.$

Thus, the correct answer is B.

2003 AMC 10B Problem 7

Answer: B

Difficulty rating: 1170

Solution:

Problem 7 in Other Years