document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209385>Question 1209385</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let a_1, a_2, a_3, ... be an arithmetic sequence.  Let S_n denote the sum of the first n terms.  If S_{10} = 1 and S_{200} = 1/2, then find S_{15}. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848733 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848733\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Write the terms of the sequence in terms of the first term a and common difference d:<br><BR>\n" );
document.write( "a, a+d, a+2d, a+3d, ...<br><BR>\n" );
document.write( "The sum of n terms of the sequence is<br><BR>\n" );
document.write( "sum = (number of terms) * (average of all terms)<br><BR>\n" );
document.write( "And in an arithmetic sequence the average of all the terms is the average of the first and last terms:<br><BR>\n" );
document.write( "sum = (number of terms) * (average of first and last terms)<br><BR>\n" );
document.write( "For the sum of the first 10 terms, the 10th term is a+9d; the sum is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%28%282a%2B9d%29%2F2%29 ALIGN=MIDDLE  ALT=\"10%28%282a%2B9d%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "For the sum of the first 200 terms, the 200th term is a+199d; the sum is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=200%28%282a%2B199d%29%2F2%29 ALIGN=MIDDLE  ALT=\"200%28%282a%2B199d%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The sum of the first 10 terms is 1, and the sum of the first 200 terms is 1/2:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%28%282a%2B9d%29%2F2%29=1 ALIGN=MIDDLE  ALT=\"10%28%282a%2B9d%29%2F2%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%2B9d=1%2F5 ALIGN=MIDDLE  ALT=\"2a%2B9d=1%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=200%28%282a%2B199d%29%2F2%29=1%2F2 ALIGN=MIDDLE  ALT=\"200%28%282a%2B199d%29%2F2%29=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a=199d=1%2F200 ALIGN=MIDDLE  ALT=\"2a=199d=1%2F200\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "This is a straightforward problem of solving a pair of linear equations in two variables, but the numbers are ugly.  Use pencil and paper if you want; or solve the pair of equations using wolframalpha.com to find<br><BR>\n" );
document.write( "first term: 7951/76000<BR>\n" );
document.write( "common difference: -39/38000<br><BR>\n" );
document.write( "Then use those to find<br><BR>\n" );
document.write( "ANSWER: the sum of the first 15 terms is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cross%281481%2F7600%29 ALIGN=MIDDLE  ALT=\"cross%281481%2F7600%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4443%2F3040 ALIGN=MIDDLE  ALT=\"4443%2F3040\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The corrected solution was verified using an excel spreadsheet:<br><BR>\n" );
document.write( "first term: 7951/76000<BR>\n" );
document.write( "common difference: -39/38000<BR>\n" );
document.write( "sum of first 10 terms: 1<BR>\n" );
document.write( "sum of first 15 terms: 4443/3040<BR>\n" );
document.write( "sum of first 200 terms: 1/2<br>\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );