Why is center frequency of a bandpass filter is given by the geometric average of the two cutoff frequencies?
Why is the center frequency of a band-pass filter is given by the geometric average of the two cutoff frequencies instead of arithmetic average?

Because its the ratios that are relevant, not the increments.

For example, if you have a bandpass filter from 2 kHz to 20 kHz, it covers a 10:1 range.  The center is then half way between these in ratio terms, which is the (square root of 10) = 3.16.  This puts the center frequency at (2 kHz)*3.16 = 6.32 kHz.  The room between the center and both ends is the same:

&nbsp;&nbsp;(20 kHz)/(6.32 kHz) = 3.2
&nbsp;&nbsp;(6.32 kHz)/(2 kHz) = 3.2