document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1087785>Question 1087785</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How much must be invested at the end of each year, for 4 years, to achieve an amount of $10000, if interest is earned at a rate of 6.25% per year, compounded annually? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #702079 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=702079\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\"><BR>\n" );
document.write( "Let P = amount invested at the end of each year<BR>\n" );
document.write( "-------------------------------<BR>\n" );
document.write( "Year 1:<BR>\n" );
document.write( "P dollars is invested at an interest rate of 6.25%, so r = 0.0625 in decimal form.<BR>\n" );
document.write( "The compounding frequency is n = 1<BR>\n" );
document.write( "The money is compounded for t = 3 years\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So,<BR>\n" );
document.write( "A = P*(1+r/n)^(n*t)<BR>\n" );
document.write( "A = P*(1+0.0625/1)^(1*3)<BR>\n" );
document.write( "A = P*1.199462890625<BR>\n" );
document.write( "A = 1.199462890625*P\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's call this A1, so A1 = 1.199462890625*P<BR>\n" );
document.write( "-------------------------------<BR>\n" );
document.write( "Year 2:<BR>\n" );
document.write( "P dollars is invested at an interest rate of 6.25%, so r = 0.0625 in decimal form.<BR>\n" );
document.write( "The compounding frequency is n = 1<BR>\n" );
document.write( "The money is compounded for t = 2 years\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So,<BR>\n" );
document.write( "A = P*(1+r/n)^(n*t)<BR>\n" );
document.write( "A = P*(1+0.0625/1)^(1*2)<BR>\n" );
document.write( "A = P*1.12890625<BR>\n" );
document.write( "A = 1.12890625*P\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's call this A2, so A2 = 1.12890625*P<BR>\n" );
document.write( "-------------------------------<BR>\n" );
document.write( "Year 3:<BR>\n" );
document.write( "P dollars is invested at an interest rate of 6.25%, so r = 0.0625 in decimal form.<BR>\n" );
document.write( "The compounding frequency is n = 1<BR>\n" );
document.write( "The money is compounded for t = 1 years\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So,<BR>\n" );
document.write( "A = P*(1+r/n)^(n*t)<BR>\n" );
document.write( "A = P*(1+0.0625/1)^(1*1)<BR>\n" );
document.write( "A = P*1.0625<BR>\n" );
document.write( "A = 1.0625*P\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's call this A3, so A3 = 1.0625*P<BR>\n" );
document.write( "-------------------------------<BR>\n" );
document.write( "Year 4:<BR>\n" );
document.write( "P dollars is invested at an interest rate of 6.25%, so r = 0.0625 in decimal form.<BR>\n" );
document.write( "The compounding frequency is n = 1<BR>\n" );
document.write( "The money is compounded for t = 0 years\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So,<BR>\n" );
document.write( "A = P*(1+r/n)^(n*t)<BR>\n" );
document.write( "A = P*(1+0.0625/1)^(1*0)<BR>\n" );
document.write( "A = P*1<BR>\n" );
document.write( "A = 1*P\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's call this A4, so A4 = 1*P<BR>\n" );
document.write( "-------------------------------\r<BR>\n" );
document.write( "\n" );
document.write( "In summary, we have the following\r<BR>\n" );
document.write( "\n" );
document.write( "A1 = 1.199462890625*P<BR>\n" );
document.write( "A2 = 1.12890625*P<BR>\n" );
document.write( "A3 = 1.0625*P<BR>\n" );
document.write( "A4 = 1*P\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Add up the values A1 through A4 to get<BR>\n" );
document.write( "A1 + A2 + A3 + A4 = 1.199462890625*P + 1.12890625*P + 1.0625*P + 1*P<BR>\n" );
document.write( "A1 + A2 + A3 + A4 = (1.199462890625 + 1.12890625 + 1.0625 + 1)*P<BR>\n" );
document.write( "A1 + A2 + A3 + A4 = 4.390869140625*P<BR>\n" );
document.write( "-------------------------------<BR>\n" );
document.write( "We want the sum of the four results (A1 through A4) to add up to $10,000 as stated in the problem.<BR>\n" );
document.write( "Set 4.390869140625*P equal to 10000 and solve for P<BR>\n" );
document.write( "4.390869140625*P = 10000<BR>\n" );
document.write( "4.390869140625*P/4.390869140625 = 10000/4.390869140625<BR>\n" );
document.write( "P = 2277.45343341673<BR>\n" );
document.write( "P = <font color=red>2277.45</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Rounded to the nearest penny, the amount you need to invest at the end of each year is <font color=red>$2,277.45</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "-------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A shortcut is to use the future value of an annuity formula<BR>\n" );
document.write( "Plug in FV = 10000, i = 0.0625 and n = 4. <BR>\n" );
document.write( "Solve for P\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "FV = P * [(1+i)^n - 1 ]/i<BR>\n" );
document.write( "10000 = P * [(1+0.0625)^4 - 1 ]/0.0625<BR>\n" );
document.write( "10000 = P * [(1.0625)^4 - 1 ]/0.0625<BR>\n" );
document.write( "10000 = P * [1.27442932128906 - 1 ]/0.0625<BR>\n" );
document.write( "10000 = P*(0.274429321289063)/0.0625<BR>\n" );
document.write( "10000 = P*4.390869140625<BR>\n" );
document.write( "10000 = 4.390869140625*P<BR>\n" );
document.write( "4.390869140625*P = 10000<BR>\n" );
document.write( "4.390869140625*P/4.390869140625 = 10000/4.390869140625<BR>\n" );
document.write( "P = 2277.45343341674<BR>\n" );
document.write( "P = <font color=red>2277.45</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Getting us the same answer as before.<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );