First slide

Functions (XII)

Question

If [x] and {x} represent the integral and fractional parts of x, respectively, then the value of $\sum_{r = 1}^{2000} \frac{\{x + r\}}{2000}$ is

Moderate

A

$\begin{array}{l} x \end{array}$
B

$\begin{array}{l} [x] \end{array}$
C

$\begin{array}{l} \{x\} \end{array}$
D

$x + 2001$

Solution

$\sum_{r = 1}^{2000} \frac{\{x + r\}}{2000} = \sum_{r = 1}^{2000} \frac{\{x\}}{2000} \sin c e i t i s f r a c t i o n a l p a r t f u n c t i o n = 2000 \frac{\{x\}}{2000} = \{x\} .$

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