document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1041797>Question 1041797</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a man borrowed 120,000 and promised to pay annually for 5 years. the payment start at \"x\" pesos. the man intends to increase his payments by 5000 at the end of each successive year. if the interest rate is 8% compounded annually, how much will the first payment be? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #657166 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=657166\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the man borrows 120,000 pesos.<BR>\n" );
document.write( "the interest rate is 8% per year compounded annually.\r<BR>\n" );
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document.write( "the present value of all his payments must be equal to 120,000.\r<BR>\n" );
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document.write( "since it's not the same payment each year, you will need to look at each year separately.\r<BR>\n" );
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document.write( "your payments at the end of each year will be:\r<BR>\n" );
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document.write( "end of year 1 = x<BR>\n" );
document.write( "end of year 2 = x + 5000<BR>\n" );
document.write( "end of year 3 = x + 10000<BR>\n" );
document.write( "end of year 4 = x + 15000<BR>\n" );
document.write( "end of year 5 = x + 20000\r<BR>\n" );
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document.write( "the present value of these payments needs to be equal to 120,000.\r<BR>\n" );
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document.write( "the present value of each of these payments will be:\r<BR>\n" );
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document.write( "present value of first payment = x / 1.08<BR>\n" );
document.write( "present value of second payment = (x + 5000) / 1.08^2<BR>\n" );
document.write( "present value of third payment = (x + 10000) / 1.08^3<BR>\n" );
document.write( "present value of fourth payment = (x + 15000) / 1.08^4<BR>\n" );
document.write( "present value of fifth payment = (x + 20000) / 1.08^5\r<BR>\n" );
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document.write( "these can be made equivalent to:\r<BR>\n" );
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document.write( "the present value of:<BR>\n" );
document.write( "x/1.08 + x/1.08^2 + x/1.08^3 + x/1.08^4 + x/1.08^5 = first part.<BR>\n" );
document.write( "plus:<BR>\n" );
document.write( "the present value of 5000/1.08^2 + 10000/1.08^3 + 15000/1.08^4 + 20000/1.08^4 = second part.\r<BR>\n" );
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document.write( "the sum of the first part and the second part must be equal to 120,000.\r<BR>\n" );
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document.write( "the second part can be calculated easily to get:\r<BR>\n" );
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document.write( "the present value of the second part = 36862.12824.\r<BR>\n" );
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document.write( "since 120,000 = the sum of the first part and the second part, then you get 120,000 = the first part + 36862.12824.\r<BR>\n" );
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document.write( "solve for the first part to get:\r<BR>\n" );
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document.write( "first part = 120,000 - 36862.12824 = 83137.87176\r<BR>\n" );
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document.write( "this means that:\r<BR>\n" );
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document.write( "83137.87176 = x/1.08 + x/1.08^2 + x/1.08^3 + x/1.08^4 + x/1.08^5\r<BR>\n" );
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document.write( "if we factor out the x, we get:\r<BR>\n" );
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document.write( "83137.87176 = x * (1/1.08 + 1/1.08^2 + 1/1.08^3 + 1/1.08^4 + 1/1.08^5)\r<BR>\n" );
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document.write( "simplify this to get:\r<BR>\n" );
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document.write( "83137.87176 = x * 3.992710037\r<BR>\n" );
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document.write( "solve for x to get x = 83137.87167 / 3.992710037 = 20822.41658\r<BR>\n" );
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document.write( "the value of x is equal to 20822.41658.\r<BR>\n" );
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document.write( "the first payment will be 20822.41658.<BR>\n" );
document.write( "the second payment will be 25822.41658.<BR>\n" );
document.write( "the third payment will be 30822.41658.<BR>\n" );
document.write( "the fourth payment will be 35822.41658.<BR>\n" );
document.write( "the fifth payment will be 40822.41658.\r<BR>\n" );
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document.write( "the present value of all these payments at 8% per year will be 120,000.\r<BR>\n" );
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document.write( "the last payment is 40822.41658.\r<BR>\n" );
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document.write( "the first payment is 20822.41658.\r<BR>\n" );
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document.write( "if you recognized that:\r<BR>\n" );
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document.write( "83137.87176 = x/1.08 + x/1.08^2 + x/1.08^3 + x/1.08^4 + x/1.08^5 is equivalent to a stream of equal payments at an interest rate of .08 per year, then you could have used an annuity calculator to solve for x.\r<BR>\n" );
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document.write( "you would have entered:\r<BR>\n" );
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document.write( "present value - 83137.87176<BR>\n" );
document.write( "future value = 0<BR>\n" );
document.write( "interest rate per year = 8%<BR>\n" );
document.write( "number of years = 5<BR>\n" );
document.write( "payments are made at the end of each period.<BR>\n" );
document.write( "the calculator would have told you that the payment is equal to 20822.42.\r<BR>\n" );
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document.write( "one such online calculator you can use can be found at this link.<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"http://arachnoid.com/finance/index.html\" target = \"_blank\">http://arachnoid.com/finance/index.html</a>\r<BR>\n" );
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document.write( "my inputs and results from the use of that calculator are shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2016/072502.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "for purposes of this analysis, disregard the sign of the result.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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