[graycat]

Above gave the calculus solution in algebra and also values from discrete arithmetic with (10)(365) steps per year.

For the arithmetic for the calculus solution, in Rexx,

     y.0 = ( 1 / 100 )
     b = 1
     k = ( 1 / 2000 )

     Do t = 1 To 5
       t1 = t * (10 * 365 )
       e1 = RxCalcExp( b * k * t1, 16 )
       y.t = ( y.0 * e1 ) /  ( y.0 * ( e1 - 1 ) + b )
       Say Format(t,9) Format(100*y.t,6,2) || '%'
     End

Table below has the values from both the calculus and the discrete versions:

         ==== Growth ====

             Customer Fraction
     Year   Calculus   Discrete
        1      5.90%      5.89%
        2     27.99%     27.97%
        3     70.68%     70.66%
        4     93.73%     93.73%
        5     98.93%     98.93%

Lesson: Sometimes in growth problems with a calculus solution, a discrete version can give close results.