document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1202908>Question 1202908</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You deposit $100 each month into an account earning 5% interest compounded monthly.<BR>\n" );
document.write( "a) How much will you have in the account in 35 years? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852257 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52803)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852257\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "You deposit $100 each month into an account earning 5% interest compounded monthly.<BR>\n" );
document.write( "How much will you have in the account in 35 years?<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;This my solution goes second after the solution by @Theo.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;@Theo used an online calculator in his solution.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Since this web-site is to teach Math, the expected solution should be mathematical.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;So, I present here a standard mathematical solution.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Calculator can be used as a secondary tool for verification.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Then the sequence of steps would be normally educative.\r<BR>\n" );
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document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "I will assume that this account is an ordinary annuity saving plan, i.e. depositing and compounding\r\n" );
document.write( "are made at the end of each month.\r\n" );
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document.write( "Use the formula for the Future Value of the ordinary annuity\r\n" );
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document.write( "    FV = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%2A%28%28%281%2Br%29%5En-1%29%2Fr%29 ALIGN=MIDDLE  ALT=\"P%2A%28%28%281%2Br%29%5En-1%29%2Fr%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >,    (1)\r\n" );
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document.write( "where  FV is the future value of the account;  P is the monthly payment (deposit); \r\n" );
document.write( "r is the monthly effective compounding rate presented as a decimal; \r\n" );
document.write( "n is the number of deposits (= the number of years multiplied by 12, in this case).\r\n" );
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document.write( "Under the given conditions, P = $100;  r = 0.05/12;  n = 35*12 = 420 payments/(compounding).\r\n" );
document.write( "So, according to the formula (1), you get for the monthly payment value\r\n" );
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document.write( "    P = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100%2A%28%28%281%2B0.05%2F12%29%5E%2835%2A12%29-1%29%2F%28%280.05%2F12%29%29%29 ALIGN=MIDDLE  ALT=\"100%2A%28%28%281%2B0.05%2F12%29%5E%2835%2A12%29-1%29%2F%28%280.05%2F12%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 113609.24  (rounded to the closest cent).\r\n" );
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document.write( "It is precisely the same value as @Theo obtained in his post using a calculator.\r\n" );
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document.write( "<U>Answer</U>.  The necessary monthly deposit value is $113609.24.\r\n" );
document.write( "</PRE>\r<BR>\n" );
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document.write( "Solved.\r<BR>\n" );
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document.write( "In such calculations, it is important to avoid intermediate rounding.<BR>\n" );
document.write( "Rounding can be done at the end, only.\r<BR>\n" );
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document.write( "Intermediate rounding is PROHIBITED, since it leads to wrong answer.\r<BR>\n" );
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document.write( "Also, a calculator should provide the necessary precision.\r<BR>\n" );
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document.write( "I use MS Excel in my computer. It works with 15 decimals in mantissa <BR>\n" );
document.write( "and provides the necessary precision.\r<BR>\n" );
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document.write( "I simply copy-paste my numerical formula into Excel spreadsheet and get the answer in one click.\r<BR>\n" );
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document.write( "Many standard online calculators provide the necessary precision.\r<BR>\n" );
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document.write( "You may use, for example, these free-of-charge online calculators\r<BR>\n" );
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document.write( "https://www.omnicalculator.com/finance/annuity-future-value\r<BR>\n" );
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document.write( "https://www.calculatorsoup.com/calculators/financial/future-value-annuity-calculator.php\r<BR>\n" );
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document.write( "https://www.calculator.net/future-value-calculator.html\r<BR>\n" );
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document.write( "They are reliable, have convenient user-friendly interface, have complete instructions and descriptions, <BR>\n" );
document.write( "provide the necessary precision and were checked million times. You may use them to check my/your calculations.\r<BR>\n" );
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document.write( "There is another (= one more) reason why I produced and placed my solution here.\r<BR>\n" );
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document.write( "@Theo' posts used visual plots to support his solutions.<BR>\n" );
document.write( "These plots were integral inseparable part of his solutions.<BR>\n" );
document.write( "But some time ago, Theo left this forum and stopped supporting web-site with his plots.<BR>\n" );
document.write( "As a result, you see now some colored spots in his posts, where his plots should be.<BR>\n" );
document.write( "Due to this reason, @Theo's posts lost their educational meaning and value.<BR>\n" );
document.write( "Therefore, I create my posts with my own mathematical solutions<BR>\n" );
document.write( "to replace @Theo' solutions and provide meaningful mathematical content.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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