document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=142024>Question 142024</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The sum of the first 12 terms of an AP (Arithmetic Progression) is 222, the sum of the first 5 terms is 40. Write out the first four terms of the series. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #103485 by </B><A HREF=/tutors/aboutme.mpl?userid=nabla><B>nabla(475)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=nabla><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=103485\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The sum of the first n terms of an arithmetic progression is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=na%2B%28n%28n-1%29r%29%2F2 ALIGN=MIDDLE  ALT=\"na%2B%28n%28n-1%29r%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Using this formula with n=12 and n=5,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=222=12a%2B%2812%2812-1%29r%29%2F2 ALIGN=MIDDLE  ALT=\"222=12a%2B%2812%2812-1%29r%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=40=5a%2B%285%285-1%29r%29%2F2 ALIGN=MIDDLE  ALT=\"40=5a%2B%285%285-1%29r%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "simplifying a bit:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=222=12a%2B66r ALIGN=MIDDLE  ALT=\"222=12a%2B66r\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=40=5a%2B10r ALIGN=MIDDLE  ALT=\"40=5a%2B10r\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "and simplifying a bit more:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=37=2a%2B11r ALIGN=MIDDLE  ALT=\"37=2a%2B11r\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8=a%2B2r ALIGN=MIDDLE  ALT=\"8=a%2B2r\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We now solve this system of equations.\r<BR>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=linear_substitution.solver>SOLVE linear system by SUBSTITUTION</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Solve:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+system%28+%0D%0A++++2%5Ca+%2B+11%5Cr+=+37%2C%0D%0A++++1%5Ca+%2B+2%5Cr+=+8+%29%0D%0A++ ALIGN=MIDDLE  ALT=\"+system%28+%0D%0A++++2%5Ca+%2B+11%5Cr+=+37%2C%0D%0A++++1%5Ca+%2B+2%5Cr+=+8+%29%0D%0A++\"   DISABLED_style= \"max-width:90%; height:auto;\" >We'll use substitution. After moving 11*r to the right, we get:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Aa+=+37+-+11%2Ar ALIGN=MIDDLE  ALT=\"2%2Aa+=+37+-+11%2Ar\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+37%2F2+-+11%2Ar%2F2 ALIGN=MIDDLE  ALT=\"a+=+37%2F2+-+11%2Ar%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Substitute that<BR>\n" );
document.write( "  into another equation:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2A%2837%2F2+-+11%2Ar%2F2%29+%2B+2%5Cr+=+8 ALIGN=MIDDLE  ALT=\"1%2A%2837%2F2+-+11%2Ar%2F2%29+%2B+2%5Cr+=+8\"   DISABLED_style= \"max-width:90%; height:auto;\" > and simplify: So, we know that r=3. Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+37%2F2+-+11%2Ar%2F2 ALIGN=MIDDLE  ALT=\"a+=+37%2F2+-+11%2Ar%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, a=2.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Answer: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28+a=2%2C+r=3+%29 ALIGN=MIDDLE  ALT=\"system%28+a=2%2C+r=3+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "a=2, r=3.\r<BR>\n" );
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document.write( "The first 4 terms of the series would be, then:<BR>\n" );
document.write( "2,(2+3),(2+2(3)),(2+3(3)) which is<BR>\n" );
document.write( "2,5,8,11\r<BR>\n" );
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document.write( "In order to check, which is always a good idea, we can only iterate:<BR>\n" );
document.write( "2+5+8+11+14=40  (first part is correct)<BR>\n" );
document.write( "2+5+8+11+14+17+20+23+26+29+32+35=222 (second part is correct). </SPAN>\n" );
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