SOLUTION: Suppose you eard 10,000 per month and pay 20% tax and spend 60% of it. Again the recipient of money spent by you pay 20% tax and spend 60% of it. If this process continues without

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Question 1099028: Suppose you eard 10,000 per month and pay 20% tax and spend 60% of it. Again the recipient of money spent by you pay 20% tax and spend 60% of it. If this process continues without end,find the total amount that will be paid for the tax
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
somehow this winds up being a geometric progression with A1 = 2000 and r = .48.

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 S_infinity = A1 / (1-r) as long as 0 < r < 1.

that's the case here, and you get S_infinity = 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.