document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1099028>Question 1099028</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 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 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #713838 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=713838\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> somehow this winds up being a geometric progression with A1 = 2000 and r = .48.\r<BR>\n" );
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document.write( "the formula for An is An = A1 * r^(n-1)\r<BR>\n" );
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document.write( "when A1 = 2000, this becomes:\r<BR>\n" );
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document.write( "A1 = 2000 * .48^(0) = 2000<BR>\n" );
document.write( "A2 = 2000 * .48^(1) = 960<BR>\n" );
document.write( "A3 = 2000 * .48^(2) = 460.8<BR>\n" );
document.write( "etc.....\r<BR>\n" );
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document.write( "the formula for Sn is Sn = A1 * (1 - r^n) / (1-r)\r<BR>\n" );
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document.write( "when n = infinity, the formula becomes S<sub>infinity</sub> = A1 / (1-r) as long as 0 < r < 1.\r<BR>\n" );
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document.write( "that's the case here, and you get S<sub>infinity</sub> = 2000 / .52 = 3846.153846.\r<BR>\n" );
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document.write( "i used excel to confirm this solution is correct.\r<BR>\n" );
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document.write( "it is.\r<BR>\n" );
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document.write( "the geometric formula was derived in the following manner.\r<BR>\n" );
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document.write( "you start with 10,000.<BR>\n" );
document.write( "20% tax on that is 2,000.<BR>\n" );
document.write( "60% of the after tax income is .6 * .8 * 10,000 = .48 * 10,000 = 4800.\r<BR>\n" );
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document.write( "the next person pays 20% tax on 4800 which is equal to 960.<BR>\n" );
document.write( "60% of the after tax income is .6 * .8 * 4800 = 2304.\r<BR>\n" );
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document.write( "the next person pays 20% tax on 2304 which is equal to 460.8<BR>\n" );
document.write( "60% of the after tax income is .6 * .8 * 2304 = 1105.92\r<BR>\n" );
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document.write( "the next person pays 20% tax on 1105.92 which is equal to 221.184\r<BR>\n" );
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document.write( "your tax progression is:\r<BR>\n" );
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document.write( "2000<BR>\n" );
document.write( "960<BR>\n" );
document.write( "460.8<BR>\n" );
document.write( "221.184<BR>\n" );
document.write( "etc.....\r<BR>\n" );
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document.write( "960 is equal to 2000 * .48<BR>\n" );
document.write( "460.8 is equal to 960 * .48<BR>\n" );
document.write( "221.184 is equal to 460.8 * .48\r<BR>\n" );
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document.write( "your common ratio is .48<BR>\n" );
document.write( "A1 is equal to 2000\r<BR>\n" );
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document.write( "that's your geometric progression.\r<BR>\n" );
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document.write( "i did confirm using excel that the sum of the tax paid is 3846.153846.\r<BR>\n" );
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document.write( "here's a printout of the excel worksheet.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2017/102911.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "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.\r<BR>\n" );
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document.write( "the tax in time period 57 is equal to 1.3545E-15 which means 1.3545 * 10^(-15) which is equivalent to .0000000000000013545.\r<BR>\n" );
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document.write( "that's an infinitesimally small number which gets even smaller the more time periods you go out.\r<BR>\n" );
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document.write( "needless to say, that doesn't add very much to the cumulative sum of the tax.\r<BR>\n" );
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document.write( "that seemed to saturate and reach a plateau somewhere around the 30th time period.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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