document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=199771>Question 199771</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello and Thank You for your help.\r<BR>\n" );
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document.write( "I know the answewr to my problem is 10 years. I kept subtracting to get the answer. I don't know how to write an equation to solve the problem. The question is\r<BR>\n" );
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document.write( "Bill will put aside $100 per month this year for retiremenmt. Next year he will put away $200 per month the third year $300 per month. Each year will increase by $100 per month. How many years will Bill save to reach $66,000? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #150139 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=150139\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You basically have the sequence: <BR>\n" );
document.write( "12*100,<BR>\n" );
document.write( "12*200,<BR>\n" );
document.write( "12*300,<BR>\n" );
document.write( "12*400,<BR>\n" );
document.write( "12*500, ...\r<BR>\n" );
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document.write( "note: I'm multiplying by 12 since he deposits money 12 times a year.\r<BR>\n" );
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document.write( "and you're adding up the terms to get: <BR>\n" );
document.write( "12*100+12*200=12*300,<BR>\n" );
document.write( "12*100+12*200+12*300=12*600,<BR>\n" );
document.write( "12*100+12*200+12*300+12*400=12*1000,<BR>\n" );
document.write( "12*100+12*200+12*300+12*400+12*500=12*1500, ...\r<BR>\n" );
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document.write( "and you want to know which partial sum will be equal to 66,000. \r<BR>\n" );
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document.write( "So you basically want to know when 12*100+12*200+12*300+12*400+12*500+...=66000\r<BR>\n" );
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document.write( "We can factor out the GCF 12*100 to get: 12*100(1+2+3+4+5+....)=66000\r<BR>\n" );
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document.write( "and we're left with a very familiar series in the parenthesis.\r<BR>\n" );
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document.write( "It turns out that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \displaystyle 1+2+3+4+5+\ldots+n=\sum_{k=1}^n k = \frac{n(n+1)}{2}\"> (this is a well known formula).\r<BR>\n" );
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document.write( "So the equation then becomes: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2A100%28%28n%28n%2B1%29%29%2F2%29=66000 ALIGN=MIDDLE  ALT=\"12%2A100%28%28n%28n%2B1%29%29%2F2%29=66000\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "where \"n\" is the number of years.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2A100%28%28n%28n%2B1%29%29%2F2%29=66000 ALIGN=MIDDLE  ALT=\"12%2A100%28%28n%28n%2B1%29%29%2F2%29=66000\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100%28%28n%28n%2B1%29%29%2F2%29=66000%2F12 ALIGN=MIDDLE  ALT=\"100%28%28n%28n%2B1%29%29%2F2%29=66000%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 12.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100%28%28n%28n%2B1%29%29%2F2%29=5500 ALIGN=MIDDLE  ALT=\"100%28%28n%28n%2B1%29%29%2F2%29=5500\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100%28n%28n%2B1%29%29%2F2=5500%2F100 ALIGN=MIDDLE  ALT=\"100%28n%28n%2B1%29%29%2F2=5500%2F100\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 100.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n%28n%2B1%29%29%2F2=55 ALIGN=MIDDLE  ALT=\"%28n%28n%2B1%29%29%2F2=55\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%28n%2B1%29=55%2A2 ALIGN=MIDDLE  ALT=\"n%28n%2B1%29=55%2A2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both sides by 2.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%28n%2B1%29=110 ALIGN=MIDDLE  ALT=\"n%28n%2B1%29=110\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5E2%2Bn=110 ALIGN=MIDDLE  ALT=\"n%5E2%2Bn=110\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5E2%2Bn-110=0 ALIGN=MIDDLE  ALT=\"n%5E2%2Bn-110=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 110 from both sides.\r<BR>\n" );
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document.write( "Notice we have a quadratic in the form of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=An%5E2%2BBn%2BC ALIGN=MIDDLE  ALT=\"An%5E2%2BBn%2BC\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=1 ALIGN=MIDDLE  ALT=\"A=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=1 ALIGN=MIDDLE  ALT=\"B=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=-110 ALIGN=MIDDLE  ALT=\"C=-110\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's use the quadratic formula to solve for \"n\":\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%28-B+%2B-+sqrt%28+B%5E2-4AC+%29%29%2F%282A%29 ALIGN=MIDDLE  ALT=\"n+=+%28-B+%2B-+sqrt%28+B%5E2-4AC+%29%29%2F%282A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the quadratic formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%28-%281%29+%2B-+sqrt%28+%281%29%5E2-4%281%29%28-110%29+%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"n+=+%28-%281%29+%2B-+sqrt%28+%281%29%5E2-4%281%29%28-110%29+%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=1 ALIGN=MIDDLE  ALT=\"A=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=1 ALIGN=MIDDLE  ALT=\"B=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=-110 ALIGN=MIDDLE  ALT=\"C=-110\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%28-1+%2B-+sqrt%28+1-4%281%29%28-110%29+%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"n+=+%28-1+%2B-+sqrt%28+1-4%281%29%28-110%29+%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%28-1+%2B-+sqrt%28+1--440+%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"n+=+%28-1+%2B-+sqrt%28+1--440+%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%281%29%28-110%29 ALIGN=MIDDLE  ALT=\"4%281%29%28-110%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-440 ALIGN=MIDDLE  ALT=\"-440\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%28-1+%2B-+sqrt%28+1%2B440+%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"n+=+%28-1+%2B-+sqrt%28+1%2B440+%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%281--440%29 ALIGN=MIDDLE  ALT=\"sqrt%281--440%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%281%2B440%29 ALIGN=MIDDLE  ALT=\"sqrt%281%2B440%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%28-1+%2B-+sqrt%28+441+%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"n+=+%28-1+%2B-+sqrt%28+441+%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=440 ALIGN=MIDDLE  ALT=\"440\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=441 ALIGN=MIDDLE  ALT=\"441\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%28-1+%2B-+sqrt%28+441+%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"n+=+%28-1+%2B-+sqrt%28+441+%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%28-1+%2B-+21%29%2F%282%29 ALIGN=MIDDLE  ALT=\"n+=+%28-1+%2B-+21%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=441 ALIGN=MIDDLE  ALT=\"441\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21 ALIGN=MIDDLE  ALT=\"21\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%28-1+%2B+21%29%2F%282%29 ALIGN=MIDDLE  ALT=\"n+=+%28-1+%2B+21%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%28-1+-+21%29%2F%282%29 ALIGN=MIDDLE  ALT=\"n+=+%28-1+-+21%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the expression. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%2820%29%2F%282%29 ALIGN=MIDDLE  ALT=\"n+=+%2820%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=++%28-22%29%2F%282%29 ALIGN=MIDDLE  ALT=\"n+=++%28-22%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+10 ALIGN=MIDDLE  ALT=\"n+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+-11 ALIGN=MIDDLE  ALT=\"n+=+-11\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify. \r<BR>\n" );
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document.write( "So the <i>possible</i> solutions are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+10 ALIGN=MIDDLE  ALT=\"n+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+-11 ALIGN=MIDDLE  ALT=\"n+=+-11\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Recall that the value of \"n\" is the number of years. Since you cannot have a negative year, this excludes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+-11 ALIGN=MIDDLE  ALT=\"n+=+-11\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the answer is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+10 ALIGN=MIDDLE  ALT=\"n+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" > which means that it will take 10 years to save up to $66,000.\r<BR>\n" );
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document.write( "If you have any questions, email me at jim_thompson5910@hotmail.com </SPAN>\n" );
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