document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1094753>Question 1094753</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The six term of an arithmetic progression is 23 and the sum of the first six terms is 78.Find<BR>\n" );
document.write( "(a)the common difference and the first term<BR>\n" );
document.write( "(b)the tenth term </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #709340 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=709340\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br>A key concept in working with arithmetic sequences and series is that you can always group the terms in pairs so that the sum in each pair is the same; or, if the number of terms in the sequence is odd, there will be a single term in the middle that is half of that common sum.<br><BR>\n" );
document.write( "So here is how I would work this problem....<br><BR>\n" );
document.write( "The sum of the first 6 terms is 78.  That means there are 3 pairs of terms, with each pair having a sum of 78/3 = 26.<br><BR>\n" );
document.write( "The 6th term is 23; it pairs up with the first term; and the sum of the first and 6th terms is 26.  That means the first term is 26-23 = 3.<br><BR>\n" );
document.write( "The 6th term, 23, is the first term, 3, plus the common difference 5 times:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=23+=+3+%2B+5d ALIGN=MIDDLE  ALT=\"23+=+3+%2B+5d\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20+=+5d ALIGN=MIDDLE  ALT=\"20+=+5d\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+4 ALIGN=MIDDLE  ALT=\"d+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "We are done with part (a): the common difference is 4; the first term is 3.<br><BR>\n" );
document.write( "Part (b) is now easy.  The 10th term is the first term, plus the common difference 9 times:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3+%2B+9%284%29+=+3%2B36+=+39 ALIGN=MIDDLE  ALT=\"3+%2B+9%284%29+=+3%2B36+=+39\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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