the formula for An is An = A1 * r^(n-1)
when A1 = 2000, this becomes:
A1 = 2000 * .48^(0) = 2000
A2 = 2000 * .48^(1) = 960
A3 = 2000 * .48^(2) = 460.8
etc.....
the formula for Sn is Sn = A1 * (1 - r^n) / (1-r)
when n = infinity, the formula becomes Sinfinity = A1 / (1-r) as long as 0 < r < 1.
that's the case here, and you get Sinfinity = 2000 / .52 = 3846.153846.
i used excel to confirm this solution is correct.
it is.
the geometric formula was derived in the following manner.
you start with 10,000.
20% tax on that is 2,000.
60% of the after tax income is .6 * .8 * 10,000 = .48 * 10,000 = 4800.
the next person pays 20% tax on 4800 which is equal to 960.
60% of the after tax income is .6 * .8 * 4800 = 2304.
the next person pays 20% tax on 2304 which is equal to 460.8
60% of the after tax income is .6 * .8 * 2304 = 1105.92
the next person pays 20% tax on 1105.92 which is equal to 221.184
your tax progression is:
2000
960
460.8
221.184
etc.....
960 is equal to 2000 * .48
460.8 is equal to 960 * .48
221.184 is equal to 460.8 * .48
your common ratio is .48
A1 is equal to 2000
that's your geometric progression.
i did confirm using excel that the sum of the tax paid is 3846.153846.
here's a printout of the excel worksheet.
you can see that the remaining balance (rembal)and the tax become very small very quickly and that the cumulative sum of the tax (cumtax) reaches a plateau very quickly as well, so that 57 time period gets you an answer that is consistent with an infinite number of time periods.
the tax in time period 57 is equal to 1.3545E-15 which means 1.3545 * 10^(-15) which is equivalent to .0000000000000013545.
that's an infinitesimally small number which gets even smaller the more time periods you go out.
needless to say, that doesn't add very much to the cumulative sum of the tax.
that seemed to saturate and reach a plateau somewhere around the 30th time period.