CRF Formula

[cyang]

It always seem to be the case that as I write the question, I start to figure out the answer.  I have just realized that the difference between the formulas seems to be related to whether the discounting/interest is applied at the beginning or end of the period, when discounting back to present value (i.e whether you discount the first year's payment or not). 

If anyone has a more formal explanation, I'm still interested. 

thanks,
Chris