I have a question regarding the loan payment formula shown below.
Calculating the Payment Amount per Period 
  The formula for calculating the payment amount is shown below.
Simple Amortization Calculation Formula
A = P X r(1 + r)n over (1 + r)n - 1
where
A = payment Amount per period
P = initial Principal (loan amount)
  r = interest rate per period
 n = total number of payments or periods
Is this formula/calculation a condensed version of a longer calculation?  I am curious to know how the (1 +r)n - 1 was developed from the longer calculation.  For example, r(1 + r)n may have been (r + rn)n.  The n's are exponents.
I thank you for whatever helpful explanation that may be provided.
Kenneth