Odd-Numbered Arithmetic Progressions: A Different Perspective

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"If we have to take three terms in an AP, it is convenient ti take them as a-d,a,a+d. In general, we take a-rd,a-(r-1)d,...,a-d,a,a+d, in case we have to take (2r+1) terms in an AP."

What do they mean by the portion in bold?
What is 'r' here ?
 
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r is an integer related to the total number of terms in the sequence (2r+1). I am not aware of the context.

The author (for some reason) is counting an arithmetic progression with an odd number of terms by starting in the middle and counting r terms in each direction. I have no idea what the point is. Usually a discussion of AP will start with the first term, while the number of terms may be odd or even.
 

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